The holders, leading 4-1 from the first leg, trailed 3-0 and were three minutes away from a shambolic exit on away goals when Roman Pavlyuchenko put them back in front on aggregate and Jermain Defoe added another.
Tottenham's 6-4 aggregate victory set up a Wembley final against Premier League, European and World club champions Manchester United who they also face this weekend in the fourth round of the FA Cup.
Redknapp looked stunned as his struggling Premier League side capitulated and admitted he thought they were "dead and buried" with some of his players lacking fight.
"One or two looked like they had a sulk on because I'm trying to bring new players in but really they should be fighting for their lives to stay at a great club like Tottenham," Redknapp, who won the FA Cup with Portsmouth last season, told Sky Sports.
"It was the most uncomfortable night of my football life. I need to do some stuff at this club to turn it around because we are a long way short of where we should be."
Championship Burnley showed more heart and desire and played better football than their top-flight opponents as they sensed a first major cup final for 47 years.
Jay Rodriguez pounced to put them 3-0 ahead in the 87th minute after Tottenham's reserve keeper Ben Alnwick dropped a cross at his feet.
TURNED PROVIDER
Robbie Blake had given Burnley the lead after 34 minutes with a cleverly disguised free kick which caught out Alnwick and Blake turned provider to set up Chris McCann for a tap-in after 72 minutes to put Burnley 2-0 up.
With away goals only counting after the extra 30 minutes, Burnley were hanging on grimly until Russian Pavlyuchenko struck to spoil what was gearing up to be a momentous night for the once great Lancashire club.
Pavlyuchenko buried a shot past Brian Jensen after one of Tottenham's few moves of real quality ended with Benoit Assou-Ekotto picking out the Russian.
"It was cruel on the players, it really was," Burnley manager Owen Coyle, whose side had seen off Fulham, Chelsea and Arsenal in this year's competition, told Sky Sports.
"It was a great finish by the lad at the end but everybody at this club can be very proud of what they've done."
NEWS: Redknapp: Mismatched side to play in FA Cup
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