Spurs included Mousa Dembele while the Toffees handed forward Cenk Tosun his debut after recently signing from Besiktas.
It was a tight start with few chances, with Son Heung-Min going close with a header and Wayne Rooney seeing a goal ruled out for offside.
The opener came for Tottenham on 26 minutes and, having caught Everton out moving the ball left to right on a number of occasions, that ploy paid off when Serge Aurier found space.
The Ivorian advanced into the box and although he scuffed what looked like a shot, Son was there to direct in. The game continued to meander and it remained 1-0 at the break.
However, it did not take long for Tottenham to change that statistic and, on 47 minutes, Kane tapped home after smart work by Son on the left.
Dele Alli then hit the side-netting and the Lilywhites continued to dominate against a Toffees side lacking any real urgency.
The brilliant Son then struck the post from the edge of the area and, just before the hour, Kane grabbed his second to become Tottenham’s record scorer in the Premier League on 98 goals.
While Kane could not miss, Son was devoid of his shooting boots and the same could be said for Alli, with the pair combining to force Jordan Pickford into a double save.
The South Korean was again involved in the fourth, playing through to Alli, who flicked on to Christian Eriksen and the Dane finished in emphatic style.
That was the end of the scoring as Spurs extended their unbeaten run against Everton to 11 games as the Toffees floundered to a sixth straight game without a win.
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