The striker, who was handed a first senior England call-up on Thursday, is now the leading Premier League goalscorer with 19, following an entertaining game at White Hart Lane
Former Leicester loanee Kane was on Thursday included in Roy Hodgson's squad for the upcoming internationals against Lithuania and Italy, and the England manager was in the stands to watch the striker's latest match-winning contribution on Saturday.
The 21-year-old put Tottenham two goals in front inside 13 minutes with a close-range finish and fortuitous goal that took a huge deflection off Robert Huth.
Leicester's last win came at Tottenham in the FA Cup back in January and they stunned the London club again as Jamie Vardy and Wes Morgan scored either side of half-time to draw Nigel Pearson's strugglers level.
Kane put the home side back in front, though, slotting home from the penalty spot to become the leading scorer in the Premier League with 19 goals and a Jeffrey Schlupp own goal restored Tottenham's two-goal cushion.
David Nugent set up a tense finale by pulling one back in the 90th minute, but Tottenham held on to push bottom side Leicester closer to an immediate return to the Championship.
Mauricio Pochettino was forced to make an early change when goalkeeper Hugo Lloris was carried off after colliding with Kyle Walker when making a last-ditch tackle on Vardy.
Michel Vorm came on to make his Premier League debut for Spurs, who were in front after only six minutes courtesy of yet another goal from Kane.
Leicester were caught out when Andros Townsend's corner was cleverly flicked on by Eric Dier and Kasper Schmeichel - making his first sta rt in 2015 - palmed the ball straight at Kane, who finished from inside the six-yard box.
The visitors posed a threat going forward, but they were two goals down after only 13 minutes in a moment to forget for Huth.
The German defender intervened to cut out Walker's cutback, but his heavy touch fell invitingly for Kane and the man of the moment's scuffed shot hit Huth and looped into the far corner of the net.
Leicester were being opened up far too easily, but Vardy gave Pearson's side a lifeline seven minutes before the break with a fine first-time finish from 12 yards after Nugent picked him out.
Nugent somehow failed to equalise from point-blank range under pressure from Ryan Mason after good work from Ulloa, but Morgan punished Tottenham for some dreadful defending to level five minutes after the break.
The Leicester captain was left unmarked from Matty James' corner and took advantage by stooping to beat Vorm with a powerful header.
Leicester looked the more likely to go on and win the game, but they were behind again after 62 minutes when Nugent was adjudged to have barged over Rose after taking a poor touch and Kane sent Schmeichel the wrong way from the spot to complete his hat-trick.
Tottenham then sealed the points with another stroke of luck when Eriksen went through and his shot was saved by Schmeichel, but hit Schlupp and rolled into the net five minutes from time.
Nugent gave the visitors a glimmer of hope in the last minute, but Tottenham clung on to bounce back from last Sunday's 3-0 defeat at Manchester United.