The impressive Son Heung-Min had opened the scoring in the first-half as he notched for a fifth straight home league match, while Christian Eriksen put the seal on an emphatic success when he finished off a wonderful move nine minutes from time.
England striker Kane rounded off a stellar 2017 by breaking Alan Shearer's record for the most Premier League goals in a calendar year and outscored Barcelona and Argentina superstar Lionel Messi for club and country over the 12-month period.
Tottenham are now fifth in the league table with 41 points from 22 matches, three points behind fourth-placed Liverpool.
That scare seemed to spark Tottenham into life, and Kane in particular began to threaten moments later with three efforts in quick succession which all needed to be saved by Pickford.
Rivals Arsenal are in sixth just two points behind Tottenham and play Bournemouth away on Sunday.
Apart from Wayne Rooney having a header ruled out for offside, at no stage did they ever look capable of doing something.
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Kane then drew another save from Pickford before the influential Son came within a whisker of deservedly doubling his personal tally when he collected Kane's crossfield pass, cut inside past two defenders and then fired a low strike against the post with the keeper beaten.
Allardyce's side are now winless in their last five Premier League games and have lost their last three.
Eriksen smashes a thumping drive into the corner to finish off a wonderful move involving Aurier, Son and Alli.
Defensive stability? Yes. Free-flowing, attacking prowess?
Spurs are in a firm three-way battle for one place in next season's Champions League and a heavy win today could see them go fourth ahead of Liverpool's clash with Manchester City.
26' - GOAL! Tottenham 1-0 Everton.