That goal saw Kane equal Sheringham's tally and the England striker broke it 12 minutes later when he prodded the ball past Jordan Pickford from close range.
Son Heung-min opened the scoring for Tottenham at Wembley with his fifth consecutive home league goal - the first Spurs player to do so since Jermain Defoe back in 2004 - before Kane doubled their lead moments after the interval.
Kane now has 98 league goals for the club after hitting his 19th and 20th of the current campaign and Pochettino was full of praise for the England worldwide.
Everton now sit ninth in the division but they have been woeful away from home having taken just 8 points from a possible 27 on the road so far this campaign.
The brilliant Son then struck the post from the edge of the area and, just before the hour, Kane grabbed his second to become Tottenham's record scorer in the Premier League on 98 goals.
Harry Kane needs two more goals to reach his Premier League century.
Rooney's late booking for an ugly chop on Jan Vertonghen encapsulated Everton's frustration and Tottenham polished off a handsome win with a stunning fourth goal, Dele Alli's back-heel teeing up Eriksen to hammer emphatically past Pickford.
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Hopefully, I'll keep scoring, said a modest Harry Kane who looks poised to scale new landmarks in the coming year after starting 2018 on a brilliant note like the previous year.
Mauricio Pochettino: "He [Harry Kane] can do everything because of his mentality and how he is as a professional".
"Of course, all the clubs and the managers want to keep their best players and we are in the same way, to keep the best and try to make the team stronger each season".
Former Spurs player, Graham Roberts, took to Twitter to express his delight at Kane's performance.
"Maybe I have to take the responsibility for playing too many attack-minded players". I saw some of the good signs of the players last week against Liverpool, and I saw some of the worst signs second half tonight.
Allardyce, who took over at Goodison Park in November following the dismissal of Ronald Koeman, has now seen his side fail to win any of their past six games in league and cup since the 3-1 win over Swansea City on 18 December.