Celtic will meet Hibernian in the Scottish Cup final after beating Dundee United in the second seven-goal Hampden thriller within 24 hours.
Kris Commons gave Celtic an 83-second lead, but Gary Mackay-Steven levelled before Jon Daly headed United in front.
Victor Wanyama headed an immediate reply and Commons chested in to put Celtic ahead after the break.
Another Daly header took the game into extra-time, during which substitute Anthony Stokes nodded in the winner.
It not only keeps alive Celtic's hopes of a league and cup double but ended a three-visit losing run at Hampden.
Interview - Celtic manager Neil Lennon
United can take heart from a performance that suggests they are becoming a thoughtful and elegant football team in recently-appointed manager Jackie McNamara's own mould.
The question was not so much how United would cope without a prolific goalscorer like Johnny Russell, whose return from a broken ankle came just too late for the striker to even appear on the bench.
United had, after all, gone eight games without defeat during his absence.
It was more how the formation adopted to compensate would handle a Celtic side for which Scotland wingers Commons and James Forrest were passed fit to start.
The answer appeared to come after just over a minute of play when, with United's three-man central midfield dragged to the right, Commons exploited the space to sidestep the desperately backtracking John Rankin and Willo Flood before blistering a 20-yard drive past goalkeeper Radoslaw Cierzniak.
Daly's second headed goal forced extra-time
Thoughts of their 6-2 thumping at the hands of Celtic in their last meeting must have crossed the United players' minds at that point, but they came back with admirable tenacity and no little skill.
Efe Ambrose blocked a Mark Millar drive off the line after Celtic goalkeeper Fraser Forster conceded an indirect free-kick inside the penalty box and Kelvin Wilson had to clear with a diving header from in front of his own goal.
United's defence was looking fragile to cross balls and Mikael Lustig should have extended Celtic's lead instead of sending a close-range header off the inside of a post after being found by a precise Emilio Izaguirre delivery.
Just as United looked like capitulating under the pressure, Daly found space after Wilson unwisely lunged for a desperate Barry Douglas clearance.
Mackay-Steven and Ryan Gauld seemed to take an age over their little triangle around the exposed Ambrose, but the former eventually curled the ball past Forster.
A fired-up United had a good claim for a penalty turned down when Wanyama barged into the back of Mackay-Steven.
United's second was not long in coming, though, as Daly bowed to flick home an inswinging Douglas free-kick.
Mackay-Steven and the goal provider appeared to switch off at the restart as the former allowed Izaguirre to cross and Wanyama rose handsomely above Douglas to level from eight yards.
United recovered to end the first half slightly on top, but - as with Hibs against Falkirk in the first semi-final - the team in green were the ones all go after the break.
Interview - Dundee United boss Jackie McNamara
Izaguirre was again the provider as the ball took a fortunate bounce off the chest of the unmarked Commons and in off the post.
Commons was then denied a hat-trick by Cierzniak's foot at the front post.
However, just as it looked like Celtic would kill the game, Mackay-Steven's cross found the head of Daly and the striker's looping header sneaked in off the far upright.
United survived a late pounding of their goal to take the game into extra-time.
A Daly header came off the post and Michael Gardyne's follow up was cleared off the line by Forster.
Forrest was denied by the goalkeeper's legs, but the winger's next act was to provide the cross from which Stokes powered in the winner.
Celtic's greater class and experience edged the game, but the real winner was football itself after two brilliant last-four ties that blew away the doom and gloom that has been surrounding the Scottish game.
Georgios Samaras produces a drilled right-footed shot from 20 yards. Save made by Radoslaw Cierzniak.
Direct free kick taken by Emilio Izaguirre.
Booking The referee shows Stuart Armstrong a yellow card for unsporting behaviour.
Unfair challenge on Georgios Samaras by Stuart Armstrong results in a free kick.
Free kick awarded for a foul by Efe Ambrose on Rory Boulding. Barry Douglas takes the direct free kick.
Free kick awarded for an unfair challenge on Gary Hooper by Willo Flood. Direct free kick taken by Fraser Forster.
Unfair challenge on Willo Flood by Georgios Samaras results in a free kick. Free kick crossed left-footed by Barry Douglas from right wing.
Free kick awarded for an unfair challenge on Emilio Izaguirre by Rory Boulding. Direct free kick taken by Fraser Forster.
Substitution Rory Boulding comes on in place of John Rankin.
The game restarts for the second half of extra time.
Assist by James Forrest.
103:01 Goal scored
Goal! - Anthony Stokes - Dundee U 3 - 4 Celtic Anthony Stokes finds the net with a headed goal from inside the six-yard box. Dundee Utd 3-4 Celtic.
Jon Daly challenges Joe Ledley unfairly and gives away a free kick. Kelvin Wilson restarts play with the free kick.
Substitution Georgios Samaras on for Kris Commons.
James Forrest takes a shot. Radoslaw Cierzniak makes a save. Joe Ledley has an effort at goal from just outside the box which goes wide of the left-hand upright.
Michael Gardyne takes a shot. Blocked by Charlie Mulgrew. Outswinging corner taken by Barry Douglas from the left by-line, Headed effort from inside the area by Jon Daly hits the post. Michael Gardyne takes a shot. Save by Fraser Forster.
Michael Gardyne takes a shot from inside the box clearing the bar. The referee blows for offside against Jon Daly. Free kick taken by Kelvin Wilson.
Joe Ledley gives away a free kick for an unfair challenge on Willo Flood. Radoslaw Cierzniak takes the direct free kick.
Michael Gardyne takes a shot. Save made by Fraser Forster.
Normal time is of over and the game goes into extra time.
Shot by Kris Commons. Radoslaw Cierzniak makes a save. Charlie Mulgrew takes a shot. Save made by Radoslaw Cierzniak. Effort from the edge of the area by Anthony Stokes goes wide of the right-hand post.
Corner taken by Kris Commons from the right by-line.
The referee blows for offside. Fraser Forster takes the free kick.
Michael Gardyne is flagged offside by the assistant referee. Fraser Forster restarts play with the free kick.
Kelvin Wilson takes the direct free kick.
Substitution (Dundee Utd) makes a substitution, with Michael Gardyne coming on for Ryan Gauld.
Free kick awarded for an unfair challenge on Beram Kayal by Ryan Gauld.
Effort on goal by Gary Mackay-Steven from 20 yards. Save by Fraser Forster.
Direct free kick taken by Barry Douglas.
Booking Beram Kayal is shown a yellow card.
Beram Kayal challenges Willo Flood unfairly and gives away a free kick.
Barry Douglas challenges Efe Ambrose unfairly and gives away a free kick. Kelvin Wilson takes the direct free kick.
Substitution Anthony Stokes replaces Mikael Lustig.
Substitution Victor Wanyama leaves the field to be replaced by Beram Kayal.
Unfair challenge on Mikael Lustig by Gary Mackay-Steven results in a free kick. Direct free kick taken by Mikael Lustig.
Jon Daly concedes a free kick for a foul on Kelvin Wilson. Fraser Forster restarts play with the free kick.
Assist on the goal came from Gary Mackay-Steven.
70:50 Goal scored
Goal! - Jon Daly - Dundee U 3 - 3 Celtic Jon Daly finds the back of the net with a headed goal from deep inside the penalty area. Dundee Utd 3-3 Celtic.
Kris Commons takes a shot. Barry Douglas gets a block in. Inswinging corner taken left-footed by Kris Commons from the right by-line, Headed effort on goal by Charlie Mulgrew from inside the penalty box misses to the right of the goal.
Fraser Forster takes the direct free kick.
Booking Willo Flood receives a caution for unsporting behaviour.
Joe Ledley fouled by Willo Flood, the ref awards a free kick.
Barry Douglas restarts play with the free kick.
Booking Victor Wanyama receives a yellow card.
Unfair challenge on Barry Douglas by Victor Wanyama results in a free kick.
Free kick awarded for a foul by Jon Daly on Kelvin Wilson. Fraser Forster takes the free kick.
Ryan Gauld produces a right-footed shot from outside the penalty box and misses right.
Victor Wanyama takes a shot from just inside the penalty box which clears the bar.
Substitution Mark Millar leaves the field to be replaced by Stuart Armstrong.
The assist for the goal came from Emilio Izaguirre.
59:47 Goal scored
Goal! - Kris Commons - Dundee U 2 - 3 Celtic A goal is scored by Kris Commons from close in to the bottom right corner of the goal. Dundee Utd 2-3 Celtic.
Charlie Mulgrew fouled by Gary Mackay-Steven, the ref awards a free kick. Kelvin Wilson takes the free kick.
A cross is delivered by Kris Commons, Headed effort from close range by Charlie Mulgrew goes wide left of the goal.
Booking Booking for Mark Millar for unsporting behaviour.
Mark Millar challenges James Forrest unfairly and gives away a free kick.
Kris Commons fouled by Willo Flood, the ref awards a free kick. Direct free kick taken by Charlie Mulgrew. Charlie Mulgrew takes a shot. Save made by Radoslaw Cierzniak.
Free kick awarded for a foul by Jon Daly on Mikael Lustig. Mikael Lustig takes the free kick.
Keith Watson concedes a free kick for a foul on Joe Ledley. The free kick is swung in left-footed by Charlie Mulgrew.
Corner from right by-line taken by Kris Commons.
Outswinging corner taken from the left by-line by Barry Douglas.
Unfair challenge on Sean Dillon by Gary Hooper results in a free kick. Free kick taken by Radoslaw Cierzniak.
Header from deep inside the area by Joe Ledley goes over the bar.
The second half begins.
45:00 +0:16 Half time
Half Time It is the end of the first-half.
Free kick awarded for an unfair challenge on Gary Mackay-Steven by Kris Commons. Gary Mackay-Steven takes the direct free kick.
Gary Mackay-Steven gives away a free kick for an unfair challenge on Kelvin Wilson. Fraser Forster takes the direct free kick.
Free kick awarded for an unfair challenge on Willo Flood by Gary Hooper. Direct free kick taken by Barry Douglas.
Barry Douglas takes the inswinging corner, clearance made by Victor Wanyama.
Free kick awarded for a foul by Victor Wanyama on Willo Flood. Free kick taken by Radoslaw Cierzniak.
John Rankin takes a shot. Blocked by Kelvin Wilson. Corner taken by Barry Douglas.
The assist for the goal came from Emilio Izaguirre.
30:50 Goal scored
Goal! - Victor Wanyama - Dundee U 2 - 2 Celtic Victor Wanyama finds the net with a headed goal from close range. Dundee Utd 2-2 Celtic.
Assist by Barry Douglas.
29:35 Goal scored
Goal! - Jon Daly - Dundee U 2 - 1 Celtic Jon Daly grabs a headed goal from close in. Dundee Utd 2-1 Celtic.
Emilio Izaguirre concedes a free kick for a foul on Gary Mackay-Steven. Centre by Barry Douglas,
A cross is delivered by Emilio Izaguirre, clearance made by Sean Dillon.
Keith Watson delivers the ball, save made by Fraser Forster.
Unfair challenge on Barry Douglas by Mikael Lustig results in a free kick. Direct free kick taken by Barry Douglas.
The assist for the goal came from Ryan Gauld.
23:10 Goal scored
Goal! - Gary Mackay-Steven - Dundee U 1 - 1 Celtic Gary Mackay-Steven gets on the score sheet with a goal from inside the box to the bottom left corner of the goal. Dundee Utd 1-1 Celtic.
John Rankin gives away a free kick for an unfair challenge on Mikael Lustig. Direct free kick taken by Mikael Lustig.
Header by Gary Hooper from deep inside the penalty area misses to the left of the goal. Header from close range by Mikael Lustig bounces off the post.
James Forrest produces a left-footed shot from just outside the penalty area that goes harmlessly over the target.
Charlie Mulgrew is adjudged to have handled the ball. The free kick is swung in left-footed by Barry Douglas, clearance by Charlie Mulgrew.
Shot from just outside the area by Charlie Mulgrew goes over the bar.
James Forrest crosses the ball, save made by Radoslaw Cierzniak.
Victor Wanyama has an effort at goal. Blocked by Willo Flood. Outswinging corner taken right-footed by James Forrest from the right by-line to the near post, Barry Douglas makes a clearance.
The ball is swung over by Mikael Lustig, clearance by Mark Millar.
Free kick awarded for a foul by Willo Flood on Kris Commons. Strike on goal comes in from Charlie Mulgrew from the free kick.
James Forrest produces a cross.
Outswinging corner taken from the left by-line by Barry Douglas, Mikael Lustig manages to make a clearance.
Direct free kick taken by John Rankin. Keith Watson takes a shot. Blocked by Efe Ambrose. Willo Flood takes a shot. Blocked by Kelvin Wilson.
Keith Watson produces a cross.
John Rankin takes a shot. Blocked by Mikael Lustig.
Victor Wanyama concedes a free kick for a foul on John Rankin. Free kick taken by Barry Douglas.
Assist by Victor Wanyama.
1:23 Goal scored
Goal! - Kris Commons - Dundee U 0 - 1 Celtic Kris Commons gets on the score sheet with a goal from outside the area to the top right corner of the goal. Dundee Utd 0-1 Celtic.
Charlie Mulgrew gives away a free kick for an unfair challenge on Keith Watson. John Souttar takes the free kick.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.