Kane claimed top spot in the scoring charts, while Tottenham fortified their hold on second place in the standings with a ninth successive home league win, moving to within seven points of leaders Chelsea.
Despite missing 10 games of the campaign through injury, Kane has amassed 24 goals in all competitions — only four behind last season's haul with more than two months remaining.
Kane had already sent Tottenham into a two-goal lead, including a spectacular long-range dipping strike, when Lukaku finally managed his first attempt on target in the 80th minute.
Premier League table
But the Belgium striker, who entered the game level with Kane on 17 league goals, did become the first away player to score at White Hart Lane in the league since December when he side-footed past Hugo Lloris.
Tottenham had been cruising but it was a nervy conclusion. Even after Dele Alli clipped Harry Winks' free kick into the net in the 90th minute, Everton immediately reduced the deficit again at the other end from a set-piece that Enner Valencia completed.
But Tottenham held on to end a nine-game unbeaten run for seventh-place Everton.