Spurs created little across the 90 minutes and were heading towards a first top-flight draw since April, only for substitute Eriksen to fire past Joe Hart in added time.
Erik Lamela and Son Heung-min squandered earlier opportunities for a lacklustre Tottenham side, but the Lilywhites finally found a way through at the death to make it 10 wins from their last 12 league games.
The victory moves third-placed Spurs three points clear of Arsenal and Chelsea, who play their games in hand on Sunday, while Burnley remain two points above the relegation zone.
After ending their eight-game winless run with victory over Brighton & Hove Albion last weekend, Burnley were happy to sit back at Wembley in an attempt to add another point to their tally.
The Clarets' switch to a five-man backline largely paid off, with Spurs restricted to just a couple of clear-cut opportunities in the first half.
Lucas Moura guided a first-time shot wide of Hart's goal after watching Lamela's 50-yard pass all the way, while Lamela was himself denied by an outrushing Hart once slotted through by Moussa Sissoko on the counter.
The first chance of the second period also fell Barnes's way, with the striker shooting straight into Kane in front of goal once picked out in the centre by former Spurs winger Aaron Lennon.
It was proving to be a frustrating afternoon for Mauricio Pochettino, who turned to matchwinner Eriksen and Son to provide some additional creativity for the final quarter of the game.
Tottenham did not create anything of note in the second half until late on, though, as Hart got down to his right to keep out Lamela's close-range shot and Son dragged a Fernando Llorente knockdown wide from five yards.
Just when it looked as though Spurs were about to suffer a hangover from their famous result at Camp Nou in midweek, Eriksen smashed the ball into the near top corner once teed up by Kane in space inside the opposition box.
TOTTENHAM HOTSPUR (4-2-3-1): Lloris; Trippier, Alderweireld, Davies, Rose; Sissoko, Skipp (Son 75'); Lamela (Llorente 82'), Lucas (Eriksen 65'), Alli; Kane
BURNLEY (5-4-1): Hart; Bardsley, Tarkowski, Mee, Long, Taylor; Lennon (Vokes 93'), Westwood, Cork, Brady (Hendrick 92'); Barnes (Wood 80')