Mauricio Pochettino's side are now unbeaten in 14 top-flight fixtures at White Hart Lane this season and produced an accomplished performance, albeit with a nervous finish, to bring Everton's own eight-game streak without defeat to an end.
Kane simply cannot stop scoring and a strike in either half was enough to take the 23-year-old to the top of the goalscoring charts while Romelu Lukaku was forced to operate on the fringes of the game, until he capitalised on Jan Vertonghen's slip late on.
Dele Alli and Enner Valencia both scored in stoppage time as part of a frantic finale but the hosts hung on.
While Kane wrote himself into the headline Lukaku was a side note and the big Belgian was tracked closely throughout and Toby Alderweireld produced a superb tackle to deny his countryman early on as the hosts started brightly.
So too did Leighton Baines when the left-back managed to get back to deny an onrushing Kyle Walker, who had latched onto Eric Dier's lofted through ball.
Then it was Eric Dier's turn to produce defensive heroics, nicking the ball away from Tom Davies as the youngster threatened to break into the penalty area.
Spurs initially struggled to settle into their normal game, with Everton seeming to sense a hesitancy among the hosts, but Kane stamped that out with a stunning long-range distance which whistled past Joel Robles and into the top corner.
The 23-year-old should have had a second moments later but could not quite get his feet set inside the penalty area before having an effort blocked, while Christian Eriksen blazed the rebound over.
Kane's strike summoned the Spurs fans from their Sunday afternoon slumber and Everton were on the ropes as Victor Wanyama saw an effort hit the post while Dele Alli fired wide before anybody realised the linesman's flag had been raised.
Ronald Koeman's side would have been glad to hear the half-time whistle but the Dutchman's words appeared to have little effect as Spurs settled into their rhythm immediately after the interval and Alli soon tested Robles with a cute header.
Vertonghen found his stride again quicker than anyone in a white shirt and slalomed his way through Everton's defence before his rising shoot was punched around the post by Robles.
Yet it was no surprise to see Kane deliver the defining blow, tucking the ball past Robles after an ever-menacing Alli had picked Morgan Schneiderlin's pocket.
Alli did well not to rise to Seamus Coleman's bait after a tangle with the Irishman as Spurs looked set to see out the game with minimal fuss.
But Vertonghen's slip gifted Lukaku an opportunity to race clear and slot past Hugo Lloris, setting up a memorable finish.
Everton's tails were up and mistakes started to creep into the hosts' game but Pochettino's men added a third through Alli's header before Valencia's effort at the death led to widespread gnawing of nails.
However, the hosts clung on to keep Chelsea's coat tails within their grasp.