The in-form England international, who took his tally in all competitions this season to 28 goals, matched Teddy Sheringham's feat after he tapped in Son Heung-min's through ball after the restart before prodding in Eric Dier's cross on the hour mark.
Midfielder Son earlier scored in his fifth straight league game to put the hosts ahead 1-0 in the 26th minute, darting in to tap home Serge Aurier's smashed cross after Christian Eriksen found the right back with a floated pass.
With Everton's defence in tatters, Eriksen completed the rout by collecting fellow midfielder Dele Alli's clever flick on before beating Everton goalkeeper Jordan Pickford with a crisp finish in the 81st minute.
Tottenham boss Mauricio Pochettino singled out Kane for praise and backed the 24-year-old to keep building on his achievements.
"He can do anything," Pochettino said. "With his mentality and his professionalism he can achieve all that he wishes. He is always looking to improve and that's what makes him special."
Fifth-placed Tottenham moved to 44 points and are level with Liverpool who take on league leaders Manchester City on Sunday.
"I'm so happy because these points give us the belief and confidence to fight with Chelsea, Liverpool and Manchester United for a place in the top four. That is our challenge, our target," Pochettino added.
(Editing by Christian Radnedge)
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