Spurs went close to breaking the deadlock on 14 minutes as Son Heung-min header sailed just over the crossbar, following a cross from Christian Eriksen.
Everton had the ball in the back of the net after 21 minutes but Wayne Rooney's strike was flagged offside.
Son was not to be denied four minutes later when the South Korean was on hand to turn home Serge Aurier's cross.
The Toffees opted for the long ball on numerous occasions as they looked to use the height of new signing Cenk Tosun.
However, the final 15 minutes of the half failed to provide any goal-mouth action as Spurs went into the break leading by one goal.
Spurs doubled their lead two minutes into the second half as Son turned provider to pick out Kane, who made no mistake from close range, 2-0.
Kane bagged his second and Spurs' third just before the hour mark by turning home Eric Dier's cross in the box.
The home sides managed to round off the scoring in the 81st minute as Dele Alli's delightful flick teed up Eriksen, who calmly slotted into the bottom corner.