The visitors raced into a 2-0 lead inside eight minutes thanks to goals from Marvin Sordell and Ross Wallace but Spurs were level before the break, hitting back through Paulinho and Etienne Capoue.
The comeback was complete after the break when Tom Heaton failed to gather Paulinho’s deep cross and Vlad Chiriches was on hand to score at the back post.
Danny Rose rounded off proceedings for the hosts, smashing home Roberto Soldado’s cross to seal a home tie with Leicester in the next round.
In what was a frantic beginning to the cup tie, Burnley made the perfect start after Sordell latched onto a 60-yard punt from Keiran Trippier and, with the Spurs defence nowhere to be seen, the striker fired a powerful volley past Michel Vorm and into the top left corner.
Things were about to get even better for the visitors as Ross Wallace’s 25-yard free kick hit Soldado and flew into the corner of the net, totally wrong-footing Vorm.
The home fans were in a state of shock but they were soon given something to cheer when Soldado flicked on Andros Townsend’s cross to Paulinho, who chested the ball down and volleyed home from five yards to give Spurs a lifeline.
The home side were starting to a grip on proceedings and went close through Benjamin Stambouli and Townsend but they should have been level just before the break.
Townsend led the Spurs counter-attack, breaking down the right and rolling the ball across to Soldado, who just had to tap the ball into the empty net from five yards out, but the Spanish striker’s scuffed shot hit the woodwork.
But that was not to be the home side’s final chance and Pochettino’s men equalised seconds before the interval when Capoue stormed onto Ben Davies’ cut back to smash the ball past the helpless Heaton.
The turnaround was complete minutes after the restart as Burnley keeper Heaton failed to claim Paulinho’s deep cross and Chiriches was on hand to bundle the ball home at the far post from close range.
Spurs had all the momentum and doubled their advantage two minutes later when Stambouli released Soldado with a cross-field pass. The striker went on to play a lovely weighted pass beyond the Burnley defenders and Rose charged in on goal to smash the ball home from three yards.
Sean Dyche’s men showed few signs of getting back into the tie, with the best opportunity coming as Sordell got away from Younes Kaboul, however the Frenchman recovered well to get a foot in with the former Bolton striker, who has now scored eight goals in 11 FA Cup appearances, poised to get an effort away on goal.
Spurs endeavoured to extend their lead as Nacer Chadli injected extra pace off the bench but they never lost control as Soldado dragged a shot wide from 20 yards and then had another easily saved by Heaton.
But it was not to matter as Spurs, after a shaky start, roared into the fourth round where they will face Leicester in an all-Premier League clash at White Hart Lane