Goals from Roman Pavlyuchenko and Jermain Defoe deep into extra time eventually sealed a Wembley date with Manchester United
on March 1 after a 6-4 aggregate win, but only after the Clarets had hauled themselves to within three minutes of pulling off the finest comeback in League Cup history.
Spurs were odds-on to finish the job after defeating their Championship opponents 4-1 in the first leg, but a timid and toothless performance from the visitors nearly put paid to their hopes of defending the trophy as they found themselves 3-0 down and lucky to reach the 90 minute mark still in the competition.
Robbie Blake's superb free-kick in the first half gave Burnley hope that they could pull off an unprecedented comeback, and hope turned to real belief with 18 minutes to go when Chris McCann swept home the hosts' second.
With Spurs visibly shaken, substitute Jay Rodriguez swept home an aggregate leveller with three minutes remaining to take the tie to extra time.
Spurs created little in the extra half-hour until Pavlyuchenko swept home the goal that put his sideback in the driving seat in the 117th minute, before Defoe grabbed second just before the final whistle.
Harry Redknapp made six changes from the side that drew with Portsmouth on Sunday including handing a debut to young goalkeeper Ben Alnwick, with first-choice stopper Heurelho Gomes ruled out with the groin injury he picked up on Spurs' last visit to the north west, a 1-0 defeat at Wigan 10 days ago.
Alnwick was called into action early on to spare the blushes of team-mate Benoit Assou Ekotto when the full-back inexplicably lofted his backpass towards his own goal and had it not been for the former Sunderland stopper leaping to tip the ball to safety, Burnley would have had the early goal they craved.
Brian Jensen diverted Defoe's drive from distance safely around the post a minute later but with Tottenham struggling to keep the ball, it was the only time during the first half that the huge keeper was called into action.
Burnley's industry and desire kept the Premier League side pegged back in their own half for much of the opening half-hour, so it was of little surprise when Coyle's men took the lead in the 33rd minute, with a wonderful free-kick from Blake.
The former Bradford and Leeds striker - looking to follow up his brilliant free-kick in the defeat at local rivals Preston on Saturday - lined up a set-piece from 25 yards out, level with the right side of the penalty area, but with Alnwick expecting a cross, Blake lifted the ball over the two-man wall, beating the Spurs debutant at his near post.
Gareth Bale, on for the injured Jamie O'Hara, spurned a great chance to kill off the tie just after the hour mark, by pulling his shot horribly wide with only Jensen to beat and in the 72nd minute, Spurs were made to regret the young Welshman's profligacy.
Blake picked the ball up on the left before teasing and torturing Chris Gunther to create the space to deliver a low cross which was turned home from close range by McCann.
Pavlyuchenko wasted another chance to make the last 15 minutes more comfortable for the visitors, side-footing wide from close range and again, Tottenham were made to pay.
With only three minutes remaining, Blake delivered a free-kick from deep into the Spurs box which Alnwick failed to claim. Substitute Jay Rodriguez was first to the loose ball, tapping home into the empty net to spark wild scenes of celebration at Turf Moor.
Jensen parried Tom Huddlestone's curling effort in the fourth minute of extra time as Tottenham went in search of the goal they needed to stay in the competition as Burnley sat back, content to defend their away-goals advantage.
The hosts could not hold on however, and Pavlyuchenko swept home the all-important Spurs goal in the 117th minute, before Defoe rubbed salt in the Clarets' wounds by beating Jensen from inside the box in the final minute of added time.
