Tottenham took the lead through James Justin’s early own goal before Kane took centre stage with two superb finishes before half-time.
It could be a hammer blow to fourth-placed Leicester’s Champions League dreams.
They are ahead of fifth placed Manchester United only on goals scored and Ole Gunnar Solskjaer’s side have played a game less than Leicester.
United return to league action against West Ham on Wednesday before a decisive showdown against Leicester at the King Power Stadium on the last day of the season next Sunday.
Chelsea, who have two games left against Liverpool and Wolves, remain one point above Leicester in third.
Brendan Rodgers’ side have won just two of their eight league games since the coronavirus hiatus, potentially squandering a golden opportunity to return to the Champions League for the first time since 2016-17.
After a fourth win in their last five games, Tottenham climb to sixth place with one game remaining at Crystal Palace.
Wolverhampton Wanderers, two points behind Tottenham, would go back above Jose Mourinho’s side if they beat Palace in their penultimate match on Monday.
Mourinho has insisted he isn’t embarrassed at having to motivate the 2019 Champions League finalists for their push to reach the less glamourous Europa League.
Mourinho’s players seem to agree judging by the way they put Leicester to the sword.
They took the lead in the sixth minute thanks to a large stroke of luck.
Son Heung-min cut back inside for a shot that took a wicked deflection off Justin as it flashed into the net, with Leicester’s appeal for offside against Kane rejected by VAR.
Leicester initially responded well and Wes Morgan’s header caused panic in the Tottenham defence, but Jamie Vardy’s backheeled effort was saved by Hugo Lloris.
Lloris was tested again moments later as he plunged to his left to make a one-handed save from Ayoze Perez’s close-range half-volley.
Kasper Schmeichel made an equally good stop at the other end when the Leicester keeper tipped over Son’s drive after he ran onto Harry Winks’ long pass.
Kane had netted 14 times in his previous 13 games against Leicester and he tormented them again in the 37th minute.
Breaking at speed from a cleared Leicester corner, Lucas Moura slipped a pass to Kane, who timed his run perfectly to beat the offside trap and guide a fine first-time finish past Schmeichel with his supposedly weaker left foot.
Kane followed his 22nd club goal of the season with an even better strike three minutes later.
Cutting in from the left side of the Leicester area, Kane shifted the ball away from the slow-to-react Ryan Bennett before curling a brilliant strike into the far corner.
Leicester were facing certain defeat, but they could still help their European ambitions by improving their goal difference.
Perez’s long-range effort was well saved by Lloris, while substitute Demarai Gray was also denied by the French World Cup winner.
Harvey Barnes couldn’t beat Lloris either as Leicester slipped to a painful defeat.