The striker, picked for England for the first time on Thursday, tapped home from close range after just six minutes before doubling the lead seven minutes later with a scuffed effort into the far corner.
Spurs – who lost goalkeeper Hugo Lloris to injury in the first minute – had been suspect defensively and conceded when Dave Nugent’s early cross was fired in by Jamie Vardy before half-time.
Wes Morgan’s header just after the break hauled the Foxes level but Kane converted from the penalty spot to restore Spurs’ lead and then Jeff Schlupp’s own goal seemed to put the game beyond doubt.
Nugent volleyed into the near corner in the dying moments to set up a tense finish but Spurs held on to restore their top-four challenge and keep Nigel Pearson’s side in deep relegation trouble.
Having selected Kane for the upcoming Euroepan Qualifier against Lithuania and subsequent friendly with Italy, Roy Hodgson watched from the stands as the 21-year-old scored his first ever league hat-trick and took his tally to 29 from 43 appearances this term.
After Lloris was stretchered off just seconds into the game following a collision with teammate Kyle Walker, Kane found his opener when Foxes keeper Kasper Schmeichel parried an Eric Dier flick into his path.
Leicester were a threat on the break, posing Walker and Tottenham’s right-hand side real problems, but it was the England right-back who helped set up Kane for his second with a low cross not cleared by Robert Huth.
Spurs should have been 3-0 up inside 22 minutes when Christian Eriksen beat three players on the edge of the box and fired against the far post, with Nacer Chadli blazing the rebound over an open goal from close range.
Pearson’s side, who knocked Tottenham out of the FA Cup at the Lane in January, were anything other than fazed and deserved to pull a goal back through Vardy after both Esteban Cambiasso and Nugent went close.
And an equaliser was equally as merited; Morgan heading home powerfully five minutes into the second half without any resemblance of a Tottenham challenge.
The visitors had chances to take the lead themselves and Huth should have done better with another free header inside the box, but Nugent was kicking himself after losing possession to Danny Rose inside his own box before pulling the defender over to concede a penalty on 64 minutes.
Kane, who had been quiet in the second half, coolly found the left-hand corner and could have had a fourth and his 30th of the season with a late effort over the bar – a tally that has not been levelled by any Spurs player since Jurgen Klinsmann in 1995.
Schlupp’s own goal proved to be the decider with Nugent volleying in Leicester’s third skilfully at the death, but Pearson’s side remain bottom of the division and without a win in eight Premier League games.
Tottenham: Lloris (6); Walker (5), Vertonghen (6), Dier (6), Rose (7); Mason (6), Bentaleb (7); Townsend (5), Eriksen (6), Chadli (5); Kane (9)
Subs: Vorm (6), Paulinho (7), Dembele (6)
Leicester: Schmeichel (6); De Laet (6), Huth (6), Morgan (7), Upson (5); James (6), Cambiasso (6), Schlupp (5), Vardy (8), Nugent (7), Ulloa (6)
Subs: Wasilewski (6), King (6), Mahrez (6)
Man of the Match: Harry Kane