Kane broke the deadlock in London and Pierre-Emile Hojbjerg doubled the lead late on as Spurs moved to within one point of leaders Arsenal.
There were chances for both sides in the first half but it remained 0-0 at the break after Everton did well to hold off a late surge from the home side.
The turning point came just before the hour mark when referee Paul Tierney ruled that Kane had been brought down by Everton goalkeeper Jordan Pickford and pointed to the spot.
Kane fired his effort into the bottom-left corner for his ninth Premier League goal of another prolific season.
Hojbjerg completed a 2-0 win in the 86th minute, condemning Frank Lampard's side to their second consecutive league defeat.
Tottenham are flying, level on 23 points with second-placed Manchester City and just one point behind north London rivals Arsenal, with City and the Gunners both in action on Sunday.
Antonio Conte's team have now won all five of their home league games this season.
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