David Robertson struck in the final minute of Wedesday's quarter-final replay to secure Dundee United a meeting with Raith Rovers in the last four of the competition.
Meanwhile, Rangers' hopes of retaining the SPL were boosted as Celtic crashed to St Mirren but a lacklustre performance in Tayside means there will be no clean sweep this season.
Rangers named an unfamiliar side, with Kevin Thomson failing with a dramatic late bid to have his suspension quashed for the clash at Tannadice.
The holders lodged an appeal against the red card he picked up in Sunday's Co-operative Insurance Cup final just a few minutes before the Scottish Football Association's deadline on Wednesday night but it was dismissed in under an hour after referee Craig Thomson opted to stand by his decision.
With Lee McCulloch also banned and Steven Davis suffering from a virus, Maurice Edu and Steven Naismith were given the nod in central midfield, with support from Kyle Lafferty and DaMarcus Beasley in the wide areas.
The front pairing also raised eyebrows, with youngsters Andrew Little and John Fleck starting and regular strike duo Kris Boyd and Kenny Miller both on the bench, as Rangers made seven changes from the side who clinched the silverware at the weekend.
United made two changes from their draw at Aberdeen, with Darren Dods and Robertson added to the side. Danny Swanson was on the bench and Andy Webster ineligible under the terms of his loan deal.
The home side started well and Jon Daly's 20-yard strike called Allan McGregor into action early on, before Craig Conway drilled narrowly wide of the Rangers goalkeeper's right-hand post.
At the other end, Fleck was allowed to embark on a surging run past several United players into the box but the end product was weak and Sean Dillon was able to clear the danger.
United were threatening again when Mihael Kovacevic - who grabbed the equaliser in the last game at Ibrox - tried his luck with an ambitious curling effort which drifted narrowly past the far upright with a full-stretch McGregor unable to get a touch.
There was another tense moment for Rangers when David Goodwillie's cut-back was cleared by Naismith and the ball trickled inches wide of his own goal.
Garry Kenneth then claimed he was held at the resultant corner by Steven Whittaker but no penalty was forthcoming from referee Dougie McDonald.
United claimed for another spot-kick just moments later - this time for Lafferty handball - but the pleas succeeded only in earning Kenneth a yellow card for his complaints.
Lafferty and Beasley were deployed up front in the second half, while Little and Fleck dropped back to midfield.
Again, United looked more dangerous in front of goal and were not too far away when a Conway shot fell into the path of Robertson whose effort appeared goal-bound until McGregor pawed to safety.
With the tie still goalless with 20 minutes to go, Rangers made their first change of the night when Fleck was withdrawn for the introduction of Nacho Novo. Lafferty then rifled just over the crossbar as the holders searched for the winner.
Rangers squandered another decent chance when Beasley whipped in a cross from the left but Little connected awkwardly in front of goal and his soft effort was comfortably gathered by goalkeeper Dusan Pernis.
Lafferty limped out of the action with 15 minutes remaining and was replaced by Miller, who grabbed the only goal of the game in Sunday's League Cup final.
The striker could have made the difference again when he released Novo out on the right before collecting the cut-back and seeing his firm shot held by Pernis.
Goodwillie had the ball in the back of the net with 87 minutes on the clock but the assistant referee had flagged for offside well before the effort crossed the line, before McGregor pulled off a great save to deny Daly.
But the Rangers goalkeeper was helpless to prevent United snatching a dramatic victory in the final minute when he raced off his line to block only for Robertson to slot past him and into the gaping net.