Kane, the Premier League's top scorer with 18 goals and December's player of the month, recently suggested that Spurs have to start winning trophies if they want to retain his services amid reported interest by Real Madrid.
For much of the match, however, Kane had played second fiddle to Son Heung-min, who broke the deadlock after 26 minutes and set up Kane's first shortly after the interval. They move onto 44 points, level with fourth-placed Liverpool, who take on Manchester City at Anfield tomorrow.
At this rate, Kane will also be targeting the most Premier League goals in one season which now stands at 31 for a 38 game season.
Yet there was little that Tosun could do about another hopeless away performance from Everton, who collapsed in the second half, and it was not his fault that he was reduced to playing second fiddle to Harry Kane.
But there was no further action in the remaining 13 minutes as the hosts went into the break a goal to the good, which quickly became two after the restart.
With 22 minutes on the clock, the visitors thought they had taken the lead when new striker Cenk Tosun headed on a corner and Wayne Rooney pounced, only for him to be flagged for offside. "Yet, today is not important because there are a lot of games to play".
In a odd game that ebbed and flowed, Wayne Rooney had the first chance after a quiet opening, with the striker latching onto a long kick from Jordan Pickford and dragging a left-footed shot wide of the right post from inside the area on 16 minutes.
England striker Kane has now scored 20 times in the Premier League this season, three more than any other player, and has hit 28 goals in 28 appearances in all competitions. In doing so, Son joined Jermain Defoe as the only Tottenham player to score a goal in five consecutive home matches.
"When the second goal went in and if it was offside it was a killer, but that doesn't allow us to throw the towel in and relinquish that defensive responsibility".
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.
On the possibility of one day surpassing Greaves, Kane added: "Let's see what happens".
"It was a fantastic performance", said Pochettino.