The Englishman, now the outright leading scorer in the English top flight, found the net either side of half time to make it nine home league wins in succession for the Lilywhites.
Romelu Lukaku did pull one back for the Toffees to become the club's outright all-time top scorer in the Premier League - the first of three late goals in a crazy conclusion to the match.
A 13th league goal of the season for Dele Alli in added time again looked to give his side some breathing space, but Enner Valencia found the net in the dying embers to add some gloss to the scoreline.
The hosts' only real sight of goal in a flat opening quarter to the match came from distance, as Ben Davies and Alli both saw attempts blocked aside from the edge of the box.
Kane had better luck 20 minutes in, however, sending the first on-target shot of the afternoon fizzing past Joel Robles from all of 30 yards out.
Robles will feel that he perhaps should have done better to get across to his right and keep the ball out, but he did make amends moments later when denying Kane from a one-on-one attempt.
Everton, on their best run of form in four years after going nine without defeat prior to today, were having to hold on and really should have been two down when Victor Wanyama's deflected effort beat Robles but came back off the post.
Spurs struggled to maintain the pressure and the final 15 minutes of the first half passed by with little incident, and it took until shortly before the hour mark for things to burst back into life.
Alli directed a header down the middle from close range and Jan Vertonghen drilled an attempt on target from the angle following a run from deep, but Spurs were rewarded moments later through a second Kane strike.
The high press paid off for the Lilywhites as Morgan Schneiderlin was caught in possession, with Kane never likely to miss when slotted through for his 14th goal in 12 competitive outings this calendar year.
It was game on once again nine minutes from time when an otherwise quiet Lukaku raced through on goal and, following a costly slip from Vertonghen, tucked the ball past Hugo Lloris to make some club history.
Kane had a chance to complete a fourth hat-trick of the year, only to see his attempted chip blocked by Robles, while Lukaku had a shot kept out by Lloris on the Frenchman's 200th appearance for Spurs.
Then came a thrilling finale to the contest, as Alli delicately flicked the ball past Robles to put the Londoners on the brink of another home triumph, before substitute Valencia swept home from close range for what proved to be a mere consolation.
Tottenham remain second in the table as a result of the landmark triumph, now just seven points behind leaders Chelsea who play their game in hand at West Ham United on Monday.