Christian Eriksen finished off a sweeping move on 81 minutes to round off a comfortable win.
Everton have won only one of their last 43 Premier League away games against one of the "big six" (1-0 at Man Utd in December 2013) - drawing 15 and losing 27.
This season has already seen Kane break Alan Shearer's record for most Premier League goals in a calendar year with 39, while also finishing 2017 with more goals than Lionel Messi and Cristiano Ronaldo.
By next summer, Eriksen will have two years left on the £70,000-a-week contract that he signed at the start of last season, and Tottenham are keen to reward the Denmark midfielder, who has five goals and six assists in the league this season, with a better deal and stave off interest from clubs overseas.
Spurs headed into the match on a 10-game unbeaten streak at home and knew that a victory would leave them just three points behind Chelsea, who recorded a goalless draw against 10-man Leicester, and level with Liverpool, who host Manchester City on Sunday.
Cenk Tosun had a forgettable debut for Everton but if he is worth £27million - after scoring goals in Turkey - you have to wonder how much Son is worth.
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The first player to reach 20 goals this season, Kane has had the most shots of his previous seasons with 6.28 per game but just one of his 20 have come from the spot.
"He can do everything because of his mentality and how he is as a professional".
Everton looked to have scored a goal that was ruled out for offside, while Harry Kane's second also looked like the forward was leaning offside.
"What I can't excuse is out capitulation after we went 2-0 down".
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.
"Maybe I have to take the responsibility for playing too many attack-minded players".
"It was a shock second-half performance". I saw some good signs of the players last week against Liverpool and I saw some of the worst signs in the second half today which is massively worrying. In this way he will break all the records in the Premier League.