Jose Mourinho's men appeared to be heading for a second successive Premier League draw after what was a largely toothless display at Turf Moor on Monday, but Son headed in from close range with 14 minutes to go after fine work from Kane.
Spurs saw more of the ball in the first period but Burnley went closer to breaking the deadlock, testing Hugo Lloris on a couple of occasions and having a goal chalked off.
James Tarkowski went closest of all for the Clarets, but his effort was cleared from right in front of the goal-line by Kane, who then brilliantly set up Son's winner with a clever header of his own for their ninth goal combination of the season already.
29 - Tottenham pair Harry Kane and Son Heung-Min have combined for 29 Premier League goals, with only Didier Drogba and Frank Lampard managing more (36). Wavelength. pic.twitter.com/d79bZuMNTh— OptaJoe (@OptaJoe) October 26, 2020
Although Spurs showed promise early on, they were unable to establish control and that encouraged Burnley, who saw an Ashley Barnes strike disallowed for offside in the 21st minute.
While that did not lead to a sustained period of pressure for Burnley, they did finish the first half in the ascendancy, with Lloris thwarting Ashley Westwood and Johann Gudmundsson.
Burnley went close again early in the second half – Josh Brownhill seizing on a wayward Ben Davies pass and driving just wide of the bottom-left corner.
Kane, who had endured a frustrating match in attack, proved decisive at the other end with 18 minutes to go, clearing Tarkowski's header off the line to keep Spurs on level terms.
Just four minutes later he was key in the Burnley area, running away from goal to head a corner delivery into Son's path and the South Korean threw himself forward to nod in the winner.
What does it mean? Unspectacular Spurs find a way
It certainly was not impressive, but Spurs got the job done eventually. Mourinho's side looked the least threatening of the two teams for much of the game, with a lot of the play suggesting Burnley might nick a result towards the end.
In the end, however, Spurs' superior individual quality got them over the line, with the Son-Kane route once again paying off for Mourinho, whose team are up to fifth.
Kane and Son deliver again
It is fair to say Spurs would be significantly worse off without Kane and Son, who have now combined for nine goals this season – one more and they will set a new club record for a single campaign.
On top of that, Kane's eighth Premier League assist of the term means he has had a hand in 13 goals in his first six matches of 2020-21, a new record for the competition.
One of Spurs' main issues until the latter stages was a lack of creativity; Lucas Moura was a part of that. He was hooked in the 57th minute as he had hardly influenced the game at all, making just one key pass. The Brazilian had 21 touches – fewer than all but two Spurs players, both of whom were substitutes – and his replacement, Erik Lamela, was much livelier.
Key Opta Facts
- Burnley have won just one of their previous 24 Premier League home games against the 'big six' sides (D4 L19), a 2-1 victory against Tottenham in February 2019.
- Tottenham have recorded clean sheets in successive games in all competitions after failing to record a shutout in 10 games stretching back to last season.
- In his managerial career, against no other side has Mourinho faced more often without losing than Burnley (10 – W6 D4), winning five of his six away matches against the Clarets.
- Kane has recorded eight assists in six Premier League appearances this season, the most he has ever provided in a single campaign in the competition.
- Spurs duo Kane and Son have combined for 29 Premier League goals, with only Didier Drogba and Frank Lampard managing more (36).