Roman Pavlyuchenko and Jermain Defoe scored extra-time goals to break Burnley hearts and hand Carling Cup holders Tottenham a mouth-watering final date with Manchester United following the 3-2 (6-4 aggregate) result.
Owen Coyle's brave Coca-Cola Championship team won the match 3-0 in normal time with goals from the outstanding Robbie Blake, Chris McCann and substitute Jay Rodriguez.
That made the tie 4-4 on aggregate following Tottenham's 4-1 first-leg win at White Hart Lane, and ensured a period of extra-time.
But just when Coyle's men looked set to progress to Wembley by virtue of having scored an away goal in the first leg of this astounding semi-final, substitute Pavlyuchenko slotted home in the 118th minute - and Defoe netted another minute later.
It was incredibly harsh on Burnley - who thoroughly exposed the frailties of Harry Redknapp's side at a freezing, windswept and rain-lashed Turf Moor.
Tottenham's Barclays Premier League status remains under threat - they are without a win in six league matches - but Redknapp is heading back to Wembley on March 1, having guided Portsmouth to FA Cup glory last season.
The Tottenham boss made six changes from the side which stumbled to a 1-1 home draw against Portsmouth on Sunday, yet none of his players made a favourable impression.
Debutant goalkeeper Ben Alnwick was at fault for Burnley's first and third goals as Coyle's men attacked with pace and vigour from the outset.
Blake, Chris Eagles and Wade Elliott were willing to support striker Martin Paterson at every opportunity .
Burnley almost took a freak lead in the sixth minute when Benoit Assou-Ekotto attempted to find Alnwick with a routine back-pass.
His touch was far too heavy and his lofted pass almost lobbed Alnwick, who made a fine one-handed save to turn the ball aside for a corner.
It was a warning to Tottenham and sparked the home crowd into life.
Yet the Clarets fans were almost silenced two minutes later when recalled winger David Bentley swung in a corner from the left.
The former Blackburn man was subjected to vicious abuse from the Burnley fans because of his Ewood Park links, and two missiles - one a toilet roll - were thrown in his direction.
Bentley remained calm, and his dangerous inswinging delivery was headed just over the bar by Jonathan Woodgate.
Of Tottenham's starting line-up, Woodgate and Didier Zokora were the only two survivors from the side which beat Chelsea 2-1 in last year's final.
McCann rifled a low shot wide of Alnwick's left-hand post in the 22nd minute, just after Bentley was booked for a cynical foul on Eagles.
On the half hour mark, a piece of delightful interplay between Eagles and Elliott on the edge of the visitors' 18-yard box gave the former a shooting opportunity that was deflected clear.
Yet the hosts continued to probe and were rewarded in style in the 34th minute when Blake claimed a brilliant opener.
Woodgate fouled Joey Gudjonsson 25 yards out and to the right of Alnwick's goal, but the rookie stopper allowed Blake to whip in a curling free-kick which beat him at his near post.
Suddenly, Turf Moor sensed the kind of upset Coyle had described beforehand when he said his side would need to ``scale Everest and beyond'' to progress to the final.
However, the pendulum swung in Tottenham's favour after the break - and Bentley combined with Defoe to provide Luka Modric with a shooting opportunity on the edge of the penalty area in the 48th minute.
The Croatia playmaker hit a left-foot shot over the bar, and in the 64th minute substitute Gareth Bale outpaced the Burnley defence down the left and drove a cross-shot hopelessly wide of the far post.
Burnley continued to probe and were rewarded in the 73rd minute when Blake showed nimble footwork to outfox Chris Gunter and fire a low cross to the far post, where McCann was lurking to side-foot the ball home.
The Clarets had Spurs on the ropes, and their fightback appeared complete when Rodriguez prodded in a loose ball after Alnwick fumbled Blake's free-kick.
Tottenham began extra-time purposefully, and Tom Huddlestone's long-range shot drew a smart stop from Jensen at his near post.
In the 118th minute, Pavlyuchenko rifled home a left-foot shot from six yards - and Defoe curled home a second moments later.