Spurs started strongly with Son Heung-min forcing a fine save from Nick Pope before Nathan Collins cleared Kane’s effort off the line.
In the eighth minute of first-half stoppage time, Spurs took the lead through a Kane penalty, his second of the week, awarded following a VAR review for a handball by Ashley Barnes.
Barnes almost equalised for Burnley on 62 minutes but his 20-yard strike hit the outside of a post.
Spurs pressed for a second but found Pope in inspired form as he made two stunning stops to deny Son.
A third win in four takes Spurs above Arsenal to fourth position on 68 points, two points ahead of their rivals having played a match more. Burnley are 17th on 34 points, level with Leeds United in 18th