Son Heung-min was the best player on the pitch as he scored the opening goal, set up Kane's first and hit the post. They move onto 44 points, level with fourth-placed Liverpool, who take on Manchester City at Anfield tomorrow.
Harry Kane needs two more goals to reach his Premier League century.
Serge Aurier was played into space on the right and his low cross into the six-yard box was tapped home by Son.
Chelsea were unable to pierce Leicester's defence and were second-best for long periods at Stamford Bridge, even after visiting left-back Ben Chilwell was sent off for a second bookable offence midway through the second half.
As Kane's scintillating form shows no signs of stopping many are of the belief that he will be the man to topple Jimmy Greaves' all-time record of 266 goals in all competitions and his manager is no different.
There was a touch of good fortune about his first, which put Tottenham 2-0 ahead, as he looked in an offside position when he converted Son's pass, but the goal was given.
Antonio Conte's side make unwanted history after latest stalemate against Leicester City
Watford 2-2 Southampton Abdoulaye Doucoure's controversial last-gasp equaliser denied Southampton a first win in 10 matches. Chelsea got the better of the Red Devils in their first Premier League meeting of 2017-18, winning 1-0 at Stamford Bridge.
Tottenham Hotspur won 4-0 against Everton in the Premier League clash on Saturday.
"Play your way back into the game and be professional about it".
Son was the architect as he went past Kenny to surge down the left flank before eluding Mason Holgate and crossing for Kane to net from close range on 47 minutes.
The England global now has 20 goals this term to leave him just 11 shy of the leading number of strikes in a 38-game Premier League season - a record held by Alan Shearer, Cristiano Ronaldo and Luis Suarez - and 14 away from Andy Cole and Shearer's joint haul of 34 during a 42-match campaign.
That was the end of the scoring as Spurs extended their unbeaten run against Everton to 11 games as the Toffees floundered to a sixth straight game without a win. "It's about how I believe, how we believe, and how we make things happen".
Let's just celebrate Kane for what he is.
Doucoure turned the ball in with his hand as he stooped for a header but, with no video assistant referees in the Premier League, the goal stood and denied Southampton a first league win in 10 games.
"It was a fantastic performance", he told BT Sport during their coverage of Saturday's late kick-off.