After Son Heung-min had given the hosts a first-half lead, the England man then added a double after the break, before Christian Eriksen added fourth to see Mauricio Pochettino's men ease home convincingly.
The 24-year-old's latest double sees Kane overtake Teddy Sheringham's goal-scoring record for Spurs', with the striker now having netted 98 goals for the North London club already.
Visitors make confident start
The hosts began to probe early against a Sam Allardyce side well-drilled in their defensive duties, with the Toffees sitting deep inviting Spurs to provide the onus to attack the Merseysiders' end.
It took almost a quarter-of-an-hour for the Lilywhites to have their first effort in earnest as Eriksen's delivery from a free-kick found the head of Son, with the South Korean's header dipping narrowly past the far post.
Everton's new man Cenk Tosun at the opposite end of the field was finding some nice touches as the Turk looked to build up an early understanding with Wayne Rooney for his new club.
The two almost combined to open the scoring mid-way through the opening period when Tosun's flick found Rooney from three yards out to nod into the Spurs net, only for the former England man to be marginally flagged for offside.
Son draws first blood
The hosts had began to warm to their task having warmed the palms of Jordan Pickford on a number of occasions before Spurs did indeed draw first blood under the Wembley arch.
Serge Aurier was occupying a increasingly attacking role down the right, and after the ball was fed to the Ivorian, his low cross-cum-shot crashed across the penalty area, Son was on hand to side-foot home from close range to score for his third consecutive game in a Spurs shirt.
The visitors was finding space at a premium, but with the high-press in place, Everton were finding joy in the wide areas, with Tosun and Gylfi Sigurdsson eager to chase balls down the flanks.
With Spurs also only having been given a handful of chances with what was still a close contest as the half-time whistle blew, Everton went into the break with plenty of encouragement.
Spurs go on the offensive
Everton needed to come out with greater drive, but within minutes of the restartTottenham doubled their lead albeit controversially.
After Son was involved again with a nice turn away from Jonjoe Kenny, as the forward bared down on goal and drifted wide to shy on goal. His left-footed shot flew toward the goal and on hand was the terrier-like Kane in customary fashion, to poke home almost on the goal-line. Replays suggested the England man was in an offside position when the pass came to him.
Spurs' second was Kane's 97th goal for the North Londoners, to draw level with club-record scorer Sheringham.
Son was proving a menace for the Toffee defence and thrashed a shot against the post just before the hour-mark, as the hosts went fully on the offensive. Indeed, it wasn't long before Kane then past the record of former striker Sheringham - in rather untidy fashion it must be said.
Eric Dier was the creator this time, who whipped an aerial ball goalward, with the clinical Kane on hand to knock the ball past Pickford to break yet a another top-flight record.
Hosts threaten rout
The third had sucked the life out of the Everton and had reduced the game to little more than an academic contest.
After a pattern of possession play from Spurs up to the final ten minutes, Tottenham then went through the gears to cut the visiting defence apart with Aurier and Son involved once more, before Dele Alli's delightful back-heel fell to Eriksen who rifled an effort past England stopper Pickford to add a fourth to the tally.
The hosts went in search of more spoils, but were happy to instead keep hold of their second clean sheet in three outings, as Spurs cantered home 4-0, with a trip to Southampton next Sunday.