Mauricio Pochettino’s side were poor in the final third as they passed up numerous chances, while Wayne Hennessey made a few good saves to keep them at bay for 89 minutes before Kane struck with a header to clinch all three points.
The visitors were denied an early penalty when Ben Davies was caught by Hennessey at the back post after getting on the end of Christian Eriksen’s free-kick and seeing the ball cleared by James Tomkins in the six-yard box.
Hennessey then pulled off a sharp save at close range to deny Harry Kane on 10 minutes after Patrick van Aanholt’s scuffed clearance went straight to the striker.
Palace’s first chance arrived two minutes later as Tomkins drew a routine save out of Hugo Lloris with a looping header from a free-kick on the left, before the French keeper was called into action again on 16 minutes to prevent Christian Benteke from getting on the end of Van Aanholt’s cross from the left.
The hosts survived another penalty appeal around the half-hour mark when Kane went down under a clumsy challenge from Tim Fosu-Mensah inside the box on the right after latching onto Dele Alli’s poke, but the Spurs youth product was flagged offside.
Van Aanholt failed to keep his half-volley down as he sliced a clearance over the crossbar from 16 yards out on 35 minutes, while the final action of the half saw Andros Townsend cut inside from the right and shoot tamely at Lloris from 21 yards out.
The north Londoners continued to struggle in front of goal after the restart, with Kane collecting Alli’s deft flick and blasting well over from 25 yards out on 48 minutes, before Eriksen skied a volley from 16 yards out three minutes later.
Kane should have broken the deadlock soon afterwards when Eriksen’s flick dropped at his feet, but the England international steered wide of the right post from just five yards out.
Serge Aurier at least tested Hennessey with a driven effort on 59 minutes after being played in on the right by Eriksen and the Palace gloveman pulled off a superb one-handed save at full stretch to keep out Davies’ shot on the turn 12 minutes later.
Hennessey had to be alert to prevent Benteke from diverting Kane’s strike into his own net at the near post on 83 minutes and the Welshman was relieved to see Aurier miss an open goal moments later when Davies’ low cross from the left reached him at the back post three yards out.
Kane was just inches away from picking out the bottom-left corner of the net with a low shot from 12 yards out that flew past the post on 86 minutes, but he finally scored in the final minute from a corner as his header at the back beat Hennessey, who could only get a glove on the ball to divert it over the heads of the two defenders on the line.