The game started in a lively fashion and Burnley came close to scoring the opening goal.
Good play from Kieran Trippier allowed him to whip in a deep cross to the back post, but Scott Arfield's volley went over the crossbar.
The encouraging start for Burnley soon dropped off and Spurs took control of the match by keeping hold of the ball.
Kane eventually gave Spurs the lead with a header from close range after getting on to the end of Nacer Chadli's cross.
Burnley grabbed an equaliser just three minutes later through Ashley Barnes, who received the ball from George Boyd on the edge of the box before smashing past Hugo Lloris and into the top corner of the net.
The exciting few minutes saw the game burst into life and Lamela added to the drama by curling in from 25 yards to give Spurs the lead at half time.
Following the restart, the visitors immediately went on the attack as Barnes fed Danny Ings. The striker was forced wide before he forced a save from Lloris.
The French keeper then had to deny Ings for a second time after the striker again combined with Barnes.
Spurs thought that they scored a third goal after 64 minutes, but the flag went up for offside and the goal was disallowed after Chadli converted Kane's pass from close range.
