The former Manchester United winger then set up Robbie Blake for a header that was only inches off target in the 23rd minute.
The Londoners mounted their first serious threat seconds later when David Bentley's corner was met by Jonathan Woodgate, and when Pavlyuchenko tried to turn it beyond Brian Jensen, the Burnley keeper made a stunning stop.
The visitors suffered a blow when influential midfield man Joey Gudjonsson limped off with a hamstring injury as Spurs finally managed to get a foothold in the game.
Redknapp sent on O'Hara for the ineffective Bentley at half-time - and the change had an instant and dramatic impact.
O'Hara's corner, two minutes after the break, was met by Dawson's powerful header for the equaliser.
And five minutes later an horrendous error from Jensen gifted the home side a second when he allowed O'Hara's volley to squirm through his grasp.
Redknapp's side had transformed the game, but Burnley's Blake was only inches off target with an excellent curling effort from 20 yards.
Paterson's persistence then caused confusion in the Tottenham penalty area, but keeper Heurelho Gomes cleared the danger.
Burnley's first-half composure had disappeared and Spurs hit the Clarets again with two more goals inside three minutes to seal the victory - and almost certainly a place in a second successive final.
Pavlyuchenko drove home the third after 65 minutes to maintain his record of scoring in every round of the competition before Michael Duff glanced O'Hara's free-kick past Jensen into his own net.
It completed an astonishing turnaround from a spell when the Championship side looked poised to spring a real shock on their Premier League opponents.
And it will now take an unlikely collapse from Spurs to give Burnley any hope of facing either Manchester United or Derby in the final.
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