Christian Eriksen wrapped up victory in the 81st with a fourth goal, smashed home from the edge of the area, after good work by Son and Dele Alli.
The England worldwide 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.
Tottenham Hotspur are now placed fifth in the league, 3 points behind the nearest Champions League spot now occupied by Liverpool.
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.
The in-form England global, who took his tally in all competitions this season to 28 goals, matched Teddy Sheringham's feat after he tapped in Son Heung-min's through ball after the restart before prodding in Eric Dier's cross on the hour mark.
Tottenham remain in fifth spot, level on 44 points and goal difference with Liverpool ahead of their game with Manchester City live on Sky Sports on Sunday, while Everton are also still in ninth place.
On a positive note, Idrissa Gueye is fit and available for selection, as is striker and latest club signing Cenk Tosun, who has arrived from Besiktas for a rumoured £27 million fee.
James Ward-Prowse scored his first two goals of the Premier League season for Southampton but that wasn't enough to give the Saints a desperately-needed win as hosts Watford rallied to draw 2-2.
"He is always thinking "improve, improve, improve and how to score goals and to improve the team" because his mentality is always collective and that is so important to explain to people".
However, the final 15 minutes of the half failed to provide any goal-mouth action as Spurs went into the break leading by one goal.
Everton had the ball in the net first but Wayne Rooney was correctly ruled offside.
"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".