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."
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.
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."
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