Kane and Lukaku began the day tied at the top of the Premier League goal-charts, and the pair took centre stage again as Kane's double put Spurs two up before Lukaku's late strike sparked hopes of a comeback.
Dele Alli restored the two-goal cushion in injury-time before substitute Enner Valencia added another goal seconds later to complete a dramatic finish at White Hart Lane.
Victory for Mauricio Pochettino's men means they reduce the gap behind league leaders Chelsea to seven points, with the Blues travelling to West Ham on Monday.
London football! Relive the action on Standard Sport's blog.
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Tottenham XI: Lloris (C), Walker, Dier, Alderweireld, Vertonghen, Davies, Wanyama, Dembele, Eriksen, Dele, Kane
Substitutes: Vorm, Trippier, Wimmer, Sissoko, Winks, Son, Janssen
Everton XI: Joel, Coleman, Baines, Williams, Funes Mori, Barry, Gana, Schneiderlin, Davies, Barkley, Lukaku
Substitutes: Stekelenburg, Jagielka, Holgate, McCarthy, Mirallas, Lookman, Valencia
WHAT THEY SAID
Mauricio Pochettino: “I want to explain that Harry [Kane] has a very good character, he is very strong, very professional.
"He is a player who always challenges you, like Dele Alli or Hugo Lloris, and that is what I like.”
Ronald Koeman: “[Harry] Kane is scoring goals but he will always score goals. In his movement, maybe Romelu [Lukaku] is physically a little stronger than Kane.
"Kane is clever in his movements. Both need support and creativity from other players behind them. Kane has a lot of support from Christian Eriksen, Dele Alli and the full-backs.”