The early weeks of the season have seen a string of unusual results and an off-colour Spurs looked likely to leave Turf Moor frustrated.
But South Korea forward Son scored his eighth league goal of the season in the 76th minute, heading in from close range after a headed pass from Kane.
Tottenham hammered Manchester United 6-1 at Old Trafford earlier this month and turned on the style as they charged into an early 3-0 lead against West Ham in their following game.
But they were pegged back by the Hammers in the closing minutes of that match and were forced to settle for a 3-3 draw.
Their attacking momentum vanished at Turf Moor, where they struggled to make an impression against Sean Dyche's side, who remain without a win this season.
Kane was able to get behind Burnley's defence in the early minutes following a Toby Alderweireld ball over the top but the England striker's first touch took him too wide and his shot sailed harmlessly over.
There was a lengthy stoppage as Alderweireld needed treatment on a bloody right eyebrow after being caught inadvertently by Ashley Barnes' flailing arm in an aerial challenge. The Belgian had his head bandaged before play continued.
Slightly against the run of play, Burnley had the ball in the back of the net when Barnes seemingly sprang the offside trap and buried the ball past Spurs goalkeeper Hugo Lloris.
However, the flag went up at the end of the move and a replay confirmed Barnes and a couple of other Burnley players had strayed a couple of yards offside.
It was not until the 38th minute that the game had its first shot on target, Ashley Westwood bringing a low save out of Lloris after Josh Brownhill had stolen the ball off Tanguy Ndombele on the edge of the area.
Johann Berg Gudmundsson then cut in from the right and forced Lloris into action once more as Burnley ended the half on top.
Tottenham again struggled to find their fluency in the early stages of the second half, with Burnley threatening to take the lead.
James Tarkowski put a header wide from close range before Mourinho threw on Erik Lamela to replace Lucas Moura in the 57th minute.
Kane came to the rescue from another Tarkowski effort, heading off the line.
Tottenham were in front in the 76th minute when Lamela's corner was flicked on by Kane and headed into the top corner at the back post by Son.
This time there was no collapse as they held on for a third win of the season.