Kane struck from 12 yards in first-half added time after referee Kevin Friend ruled that the ball flicked Ashley Barnes’ outstretched arm after a VAR check.
That proved enough for the vital three points which sees Spurs climb above their north London rivals into fourth, ahead of the Gunners’ trip to Newcastle on Monday night.
It was a nervy 90 minutes at the Tottenham Hotspur Stadium, that would have been much less stressful had Burnley goalkeeper Nick Pope not made a string of fine saves.
Burnley almost levelled in the second half when Barnes hit the post from distance but this defeat kept the pressure on the Clarets in the relegation fight.
It was always going to be a game of attack versus defence and Spurs were camped in Burnley’s half for the opening 20 minutes.
An early strike would have alleviated the pressure and changed the dynamic of the match but Spurs missed some chances to make it easier for themselves.
Kane headed straight at Pope after Lucas Moura’s cross while Pope also had to scramble to palm away a Son effort at the near post.
A big moment came in the 16th minute when Ryan Sessegnon floated in a cross for Kane, but the England captain’s header back across goal was cleared off the line by Nathan Collins.
The edginess around the stadium began to increase as Burnley enjoyed a rare spell of possession and they created the best chance of the game in the 28th minute.
Josh Brownhill advanced down the left and slipped in Maxwel Cornet, but the Ivorian’s shot was well saved by Hugo Lloris.
Spurs were being frustrated as Burnley held firm and made things difficult before a crucial passage of play in first-half injury time.
The hosts forced a corner which the Clarets appeared to clear before Moura danced into the penalty area and pulled back for Kane, who rifled a shot wide.
However, despite minimal appeals at the time, VAR reviewed a possible handball when the corner was cleared.
After viewing the pitchside monitor, Friend ruled that the ball had flicked the outstretched arm of Barnes and awarded a penalty.
It was a lifeline for Spurs and Kane grabbed it with both hands, sending a low penalty into the corner, with Pope rooted to the spot.
Burnley did not take the decision well, surrounding Friend after the goal went in and the benches clashed in the tunnel at half-time.
The visitors used their frustration in a positive way as they started the second half with vigour and intent.
Kevin Long headed a corner just wide before they came inches away from equalising just after the hour as Dwight McNeil fed Barnes, whose 25-yard effort crashed into the post.
Burnley pushing bodies further up the pitch did open them up to the counter attack and Spurs enjoyed more space in attack.
And they almost took advantage only to be denied by a wonder save from Pope.
They switched the ball from right to left with some neat one-touch passing, ending with Sessegnon pulling back for Son, who seemed primed to score with a low shot until Pope stuck a hand out and turned it around the corner.
He was at it again in the 81st minute as the same pair combined again, this time to bat away another effort that was destined for the roof of the net.
Those saves made it a nervy ending for Spurs, who clung on to deliver a huge three points.