Dundee Utd 2-2 Falkirk (4-5 pens)
Fighting Falkirk turned in a perfect penalty shoot-out display to reach the League Cup semi-finals at the expense of wasteful Dundee United.
The First Division took the lead thanks to a 70th-minute header from Farid El Alagui but Johnny Russell lashed in a powerful shot to level immediately.
Jon Daly knocked in a smart finish in extra time but Ally Graham hit back for the visitors in the 118th minute.
United's Willo Flood was the only man to miss in the spot-kick shoot-out.
Stewart Murdoch kept his cool with the final penalty, prompting wild scenes of celebration from the side that eliminated last season's winners Rangers in the previous round.
United, who are now on a five-match winless run, spurned a number of good chances, with Scott Robertson heading wide after 11 minutes before El Alagui's glancing header was held by home keeper Dusan Pernis.
Lauri Dalla Valle's header had Michael McGovern back-pedalling but the Falkirk goalkeeper tipped the ball onto the bar and out for a corner.
And the Finnish striker diverted a Flood drive wide before Daly latched onto John Rankin's pass, smacking a left-foot volley from the edge of the area off the crossbar.
Daly and Barry Douglas both wasted headers before the Bairns then went ahead against the run of play.
Substitute Dale Fulton's cross was met by a smart El Alagui header which beat Pernis for the striker's 15th goal in 16 games.
The lead was shortlived, however, as Russell's persistence down the left side of the box paid off when he drilled a shot low across goal and into the far corner.
Russell had two further opportunities to find a winner in normal time, but the match went to extra-time.
United went in front after 96 minutes, with Russell the provider, playing the ball through for Daly to shoot in across goal from the right side of the area.
Daly hit the woodwork for a second time when his 25-yard volley from Pernis' long kick crashed onto the bar, with McGovern well beaten.
El Alagui shot across the face of goal from a narrow angle before Daly headed just wide from a Flood corner.
McGovern then saved a Russell effort from 18 yards and Garry Kenneth hit a free-kick off target.
Pernis saved El Alagui's point-blank effort four minutes from time but, soon after, McGovern's clearance evaded the United defence and Graham, who was introduced after 110 minutes, stole in to prod past Pernis.
McGovern then pulled off a great save to claw Gavin Gunning's close-range effort round the post and the tie went to penalties.
After two successful kicks for each side, Flood ballooned his effort over and Murdoch completed a 100% success rate from five efforts for the Bairns to progress.
Dundee United Peter Houston: "In the first half, we created numerous chances and didn't take them and in the second half as well.
"Then we went behind and the players showed great character to get back into it quickly and go ahead.
"The missed chances have cost us and a long punt up the park and what looked like a foul against us with a couple of minutes to go and it's 2-2.
"Good luck to Falkirk, I am very disappointed as it was a game we should have won out the park as we created enough chances to win three matches."
Falkirk director of football Alex Smith: "Steven Pressley has the ability and the temperament to be the manager of a big club. He's got a big future in front of him.
"He's now doing the business for Falkirk Football Club, which is fantastic.
"We're going to Hampden, but we have a chance to get to the final. Let's see the draw first."