Tom Huddlestone and Emmanuel Adebayor missed as Tottenham lost a penalty shoot-out at Basel to end their hopes of winning the Europa League.
Basel won 4-1 on penalties after the quarter-final second leg ended 2-2 after extra-time, 4-4 on aggregate.
Clint Dempsey had put Spurs ahead before Mohamed Salah equalised.
Aleksandar Dragovic made it 2-1 from close range before Dempsey levelled, while Spurs' Jan Vertonghen was sent off close to the end of normal time.
Basel will join Chelsea, Fenerbahce and Benfica in Friday's semi-final draw after Huddlestone saw his penalty saved and Adebayor kicked his disappointingly over the crossbar.
It was an agonising way for Spurs to bow out of Europe.
Boss Andre Villas-Boas will point to the absence of injured trio Gareth Bale, Aaron Lennon and William Gallas.
But Tottenham, facing a tense fight to secure a top-four Premier League finish and a place in the next season's Champions League, were architects of their own downfall as they made errors at key stages.
Did Emmanuel Adebayor ever look like he had the confidence of scoring? It was an absolutely woeful penalty. It looked like he changed his mind halfway through the run up. The ball soared over the crossbar and it summed up his performance.
Having not won away from home in Europe since December 2011, they could not have wished for a better start after Dragovic's terrible error on a rain-soaked night in Switzerland.
Dragovic looked certain to cut out Vertonghen's 23rd minute through-ball but instead slipped on the greasy surface, allowing Dempsey to skip away from keeper Yann Sommer and roll the ball into the net.
However, the lead lasted no more than four minutes.
Egyptian winger Salah, who caused the Spurs defence countless problems in the first leg, found the bottom corner with a left foot shot after the visitors had carelessly squandered possession in midfield.
Mousa Dembele forced a fine save from Yann Sommer before the Swiss league leaders took the lead - for the first time over the two legs - four minutes into the second half.
Defender Dragovic, who gifted Spurs their opener, scored from close range after Friedel parried Fabian Schar's header into his path.
How the penalty shoot-out developed
1-0 to Basel - Fabian Schar
1-0 - Tom Huddlestone miss
2-0 - Marco Streller
2-1 - Gylfi Sigurdsson
3-1 - Fabian Frei
3-1 - Emmanuel Adebayor miss
4-1 - Marco Diaz
In an absorbing contest, Dragovic should have made it 3-1 in the 57th minute from a header before Dempsey forced extra-time with a goal against the run of play eight minutes from the end of normal time.
The former Fulham player brought the ball down on his chest after a hopeful ball forward before finding the net with a left-footed shot.
But Tottenham's joy was once again short-lived, Vertonghen deservedly handed his marching orders after chopping down Streller.
Mohamed Elneny's long range attempt rattled the outside of the post in the 98th minute, but Basel struggled to make the extra man advantage count.
Schar set the tone for Basel's penalty shootout success by beating Brad Friedel for pace and power.
Sommer brilliantly kept out Huddlestone's attempt but, after Marco Streller and Fabian Frei had also scored in the shootout, the Basel keeper was not needed when Adebayor skied his kick high and wide.
Gylfi Sigurdsson did manage to find the net but Marco Diaz's attempt, which made it 4-1, sparked wild scenes of celebration among the home fans.
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Live Text Commentary
Marcelo Diaz scores a penalty. Basle 4-1 Tottenham on penalties.123:29
Spot kick missed by Emmanuel Adebayor.122:55 Penalty
Penalty scored by Fabian Frei. Basle 3-1 Tottenham on penalties.122:28 Penalty
Penalty scored by Gylfi Sigurdsson. Basle 2-1 Tottenham on penalties.121:51 Penalty
Penalty scored by Marco Streller. Basle 2-0 Tottenham on penalties.121:15
A save by Yann Sommer from penalty by Tom Huddlestone.120:06 Penalty
Fabian Schar scores a penalty. Basle 1-0 Tottenham on penalties.120:01
All is set for the penalty shoot-out to start.120:00 +1:24
The match goes to a penalty shoot-out.120:00 +0:01
Markus Steinhofer challenges Kyle Walker unfairly and gives away a free kick. Brad Friedel restarts play with the free kick.119:35
Alexander Frei delivers the ball, save by Brad Friedel.119:06
Foul by Tom Carroll on Philipp Degen, free kick awarded. Free kick taken by Fabian Frei.118:04
Corner from the left by-line taken by Alexander Frei, clearance by Tom Huddlestone.115:13
Shot from just outside the box by Mohamed El-Nenny goes over the bar.113:55
The ball is delivered by Markus Steinhofer, fantastic save by Brad Friedel.112:13
Outswinging corner taken from the left by-line by Alexander Frei, Foul by Philipp Degen on Brad Friedel, free kick awarded. Brad Friedel takes the direct free kick.111:59
Alexander Frei takes a shot. Blocked by Michael Dawson.110:35 Substitution
Substitution (Basle) makes a substitution, with Alexander Frei coming on for Mohamed Salah.105:45
Clint Dempsey fouled by Fabian Frei, the ref awards a free kick. Direct free kick taken by Kyle Walker.105:01
The match restarts for the second half of extra time.
Marcelo Diaz produces a left-footed shot from just outside the penalty box that goes wide left of the target.102:35
Centre by Tom Huddlestone, Michael Dawson takes a shot. Yann Sommer makes a save.102:35 Booking
Booking Mohamed El-Nenny goes into the book for unsporting behaviour.102:24
Free kick awarded for a foul by Aleksander Dragovic on Emmanuel Adebayor.101:41
Emmanuel Adebayor concedes a free kick for a foul on Fabian Schar. Direct free kick taken by Aleksander Dragovic.100:48
The assistant referee signals for offside against Marco Streller. Brad Friedel restarts play with the free kick.98:37
Gylfi Sigurdsson takes the outswinging corner, Philipp Degen manages to make a clearance.97:22
Mohamed El-Nenny has an effort at goal from a long way out hitting the post.96:18
Fabian Schar challenges Michael Dawson unfairly and gives away a free kick. Free kick taken by Brad Friedel.95:06
Effort on goal by Joo-Ho Park from 18 yards. Brad Friedel makes a save.93:31
Free kick awarded for an unfair challenge on Clint Dempsey by Joo-Ho Park. Brad Friedel restarts play with the free kick.92:49
Unfair challenge on Marcelo Diaz by Tom Huddlestone results in a free kick. Markus Steinhofer delivers the ball from the free kick right-footed from right channel, Michael Dawson makes a clearance.91:46
The assistant referee flags for offside against Philipp Degen. Brad Friedel takes the free kick.90:00 +3:15
Aleksander Dragovic has a direct shot on goal from the free kick.90:00 +3:15 Booking
Booking Kyle Walker goes into the book for unsporting behaviour.90:00 +3:10 Booking
Booking Marco Streller booked for unsporting behaviour.90:00 +2:30 Booking
Booking Michael Dawson receives a yellow card.90:00 +2:27
Michael Dawson gives away a free kick for an unfair challenge on Aleksander Dragovic.89:27
Shot comes in from Marcelo Diaz from the free kick.89:27 Dismissal
Dismissal The referee shows Jan Vertonghen a red card.89:21
Unfair challenge on Marco Streller by Jan Vertonghen results in a free kick.87:47
The official flags Emmanuel Adebayor offside. Free kick taken by Aleksander Dragovic.86:45
Corner taken right-footed by Gylfi Sigurdsson from the left by-line, save by Yann Sommer.84:36
Emmanuel Adebayor concedes a free kick for a foul on Joo-Ho Park. Yann Sommer takes the free kick.83:47
The ball is sent over by Markus Steinhofer, Marco Streller takes a shot. Save made by Brad Friedel.81:30
Assist on the goal came from Tom Huddlestone.81:30 Goal scored
Goal! - Clint Dempsey - FC Basel 2 - 2 Tottenham Clint Dempsey fires in a goal from inside the six-yard box to the bottom right corner of the goal. Basle 2 (4)-(4) 2 Tottenham.81:01
Marcelo Diaz gives away a free kick for an unfair challenge on Gylfi Sigurdsson. Free kick taken by Tom Huddlestone.80:07
Effort from just outside the box by Gylfi Sigurdsson goes over the bar.77:53 Substitution
Substitution Benoit Assou-Ekotto comes on in place of Kyle Naughton.76:43 Substitution
Substitution Tom Huddlestone on for Scott Parker.76:28
Marcelo Diaz takes a shot. Save by Brad Friedel.73:12
Jan Vertonghen challenges Yann Sommer unfairly and gives away a free kick. Yann Sommer takes the direct free kick.72:14
Scott Parker takes a shot. Blocked by Fabian Schar. Corner taken right-footed by Gylfi Sigurdsson from the right by-line, Michael Dawson takes a shot. Save made by Yann Sommer.70:41
The offside flag is raised against Clint Dempsey. Yann Sommer restarts play with the free kick.69:14 Substitution
Substitution Markus Steinhofer joins the action as a substitute, replacing Valentin Stocker.68:18
The official flags Emmanuel Adebayor offside. Yann Sommer restarts play with the free kick.64:48
Mohamed El-Nenny takes a shot. Brad Friedel makes a save.63:42
Philipp Degen gives away a free kick for an unfair challenge on Gylfi Sigurdsson. Lewis Holtby restarts play with the free kick.62:58
Philipp Degen has an effort at goal from deep inside the area missing to the right of the target.61:15
The offside flag is raised against Marco Streller. Jan Vertonghen takes the indirect free kick.60:33
Mohamed El-Nenny fouled by Gylfi Sigurdsson, the ref awards a free kick. Yann Sommer takes the direct free kick.59:23
Gylfi Sigurdsson is caught offside. Yann Sommer restarts play with the free kick.57:34
Free kick taken by Scott Parker.57:34 Substitution
Substitution Tom Carroll comes on in place of Mousa Dembele.57:34 Substitution
Substitution Marcelo Diaz on for Serey Die.57:34 Booking
Booking Booking for Serey Die.57:28
Scott Parker fouled by Serey Die, the ref awards a free kick.56:01
Free kick awarded for a foul by Michael Dawson on Aleksander Dragovic. Mohamed El-Nenny sends in a cross, Aleksander Dragovic has a headed effort at goal from deep inside the area missing to the right of the goal.55:10
Kyle Naughton takes a shot. Blocked by Fabian Schar.54:44
Corner taken by Gylfi Sigurdsson from the left by-line, save by Yann Sommer.53:43
Effort from inside the six-yard box by Emmanuel Adebayor goes wide left of the target.52:50
Free kick awarded for a foul by Fabian Frei on Jan Vertonghen. Jan Vertonghen takes the direct free kick.52:01 Booking
Booking Clint Dempsey goes into the book for dissent.51:20
Direct effort from the free kick comes in from Gylfi Sigurdsson.51:12
Clint Dempsey fouled by Valentin Stocker, the ref awards a free kick.48:18 Goal scored
Goal! - Aleksander Dragovic - FC Basel 2 - 1 Tottenham Goal scored by Aleksander Dragovic from close in low into the middle of the goal. Basle 2 (4)-(3) 1 Tottenham.48:17
Valentin Stocker takes a inswinging corner to the near post.47:09
Inswinging corner taken from the left by-line by Gylfi Sigurdsson, Emmanuel Adebayor takes a shot. Comfortable save by Yann Sommer.45:01
The referee blows his whistle to start the second half.45:00 +2:02 Half time
Half Time The half-time whistle blows.45:00 +1:48
Outswinging corner taken by Valentin Stocker.44:59
Fabian Frei takes the direct free kick. Fabian Frei takes a shot.44:59 Booking
Booking Mousa Dembele is cautioned by the ref for unsporting behaviour.44:51
Mousa Dembele challenges Valentin Stocker unfairly and gives away a free kick.44:33
Kyle Naughton takes a shot. Save by Yann Sommer.43:38
A cross is delivered by Marco Streller.41:26
Michael Dawson fouled by Valentin Stocker, the ref awards a free kick. Brad Friedel takes the free kick.38:48
The free kick is delivered left-footed by Mohamed Salah from right wing, Michael Dawson makes a clearance.38:48 Booking
Booking Kyle Naughton is shown a yellow card.38:46
Kyle Naughton challenges Mohamed Salah unfairly and gives away a free kick.36:04
Mousa Dembele takes a shot. Yann Sommer makes a fantastic save. Emmanuel Adebayor takes a shot. Save by Yann Sommer.32:48
Gylfi Sigurdsson fouled by Philipp Degen, the ref awards a free kick. Free kick taken by Kyle Naughton.31:26
The ball is swung over by Marco Streller, clearance by Michael Dawson.30:21
Outswinging corner taken right-footed by Gylfi Sigurdsson from the right by-line, Clint Dempsey takes a shot. Save by Yann Sommer.28:09
The assistant referee flags for offside against Emmanuel Adebayor. Yann Sommer restarts play with the free kick.26:32
Assist on the goal came from Marco Streller.26:32 Goal scored
Goal! - Mohamed Salah - FC Basel 1 - 1 Tottenham Mohamed Salah finds the back of the net with a goal from the edge of the area low into the middle of the goal. Basle 1 (3)-(3) 1 Tottenham.25:48
Clint Dempsey gives away a free kick for an unfair challenge on Fabian Schar. Fabian Frei restarts play with the free kick.22:30
Jan Vertonghen provided the assist for the goal.22:30 Goal scored
Goal! - Clint Dempsey - FC Basel 0 - 1 Tottenham Goal scored by Clint Dempsey from inside the penalty area low into the middle of the goal. Basle 0 (2)-(3) 1 Tottenham.19:06
Free kick awarded for an unfair challenge on Fabian Schar by Emmanuel Adebayor. Free kick taken by Fabian Schar.18:38
The offside flag is raised against Valentin Stocker. Free kick taken by Brad Friedel.17:14
Mohamed Salah fouled by Lewis Holtby, the ref awards a free kick. Philipp Degen takes the direct free kick.15:26
Shot from long range by Gylfi Sigurdsson misses to the right of the goal.14:10
Michael Dawson gives away a free kick for an unfair challenge on Fabian Schar. Fabian Frei takes the direct free kick.13:32
Emmanuel Adebayor is flagged offside by the assistant referee. Yann Sommer restarts play with the free kick.11:46
Effort on goal by Mohamed Salah from deep inside the area goes harmlessly over the bar.9:11
Clint Dempsey has an effort at goal from outside the area which goes wide left of the target.7:33
Foul by Aleksander Dragovic on Kyle Walker, free kick awarded. Jan Vertonghen produces a shot on goal direct from the free kick.1:55
Scott Parker challenges Mohamed Salah unfairly and gives away a free kick. Direct strike on goal from the free kick comes in from Fabian Schar.1:00
Philipp Degen gives away a free kick for an unfair challenge on Gylfi Sigurdsson. Direct free kick taken by Gylfi Sigurdsson.0:00
The referee gets the match started.
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