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For a while, it looked as if Kane might be outshone by his team mate Son Heung-min, who scored in a home game for the fifth successive time.
Then Kane took over to net in the 47th and 59th minutes to take his season's league tally to 20, the fourth consecutive campaign he has reached the landmark.
Christian Eriksen rounded off a cleverly worked team move, again featuring Son, for Spurs' fourth as Everton crashed to their heaviest defeat under Sam Allardyce, who gave a debut to 27 million pound (NZ$51 million) new signing Cenk Tosun.
With three of the top four not playing, fifth-placed Tottenham moved level with Liverpool on 44 points, three behind Chelsea. Antonio Conte's side missed the chance to go second as Leicester hung on for a draw at Stamford Bridge despite the 68th-minute dismissal of Ben Chilwell.
It was the third successive game in all competitions that Chelsea failed to score, with striker Alvaro Morata looking frustrated and short of confidence.
Manager Conte believes they are paying the price for playing seven games in three weeks, with no let-up in prospect as they contest four competitions. "I think I saw a lot of very tired players," Conte said.
Elsewhere, there were more controversial refereeing decisions as English football continued to debate whether the video assistant referee (VAR) system, which was tested in some cup games last week, would benefit the Premier League.
The first flashpoint occurred at Vicarage Road where TV replays revealed Watford's Abdoulaye Doucoure bundled the ball over the line with his right hand in the 2-2 draw with Southampton.
A seventh Premier League defeat in eight games had looked likely for Marco Silva's side after James Ward-Prowse scored twice but Andre Gray set up a thrilling finale before Doucoure's contribution led to furious Southampton protests.
"We can do nothing now," said Southampton manager Mauricio Pellegrino diplomatically. However, Watford striker Troy Deeney admitted that had the VAR system been in operation "we probably wouldn't have got it".
At Newcastle United, bottom side Swansea City believed they were unlucky not to be awarded a penalty when Mohamed Diame appeared to handle the ball.
Swansea manager Carlos Carvalhal said: My assistant says that it's a very clear penalty and a red card also, but what can we do? Nothing, only talk about this. This is one of the situations VAR will in the future say clear penalty – it will be clear."
There was nothing controversial about West Ham United's 4-1 destruction of Huddersfield Town, a victory that puts the Hammers 11th in the table.
When David Moyes took over as West Ham manager many observers questioned his ability to get the team scoring again. But the Scot has proved all the doubters wrong with the London side scoring 12 goals from his past five league games in charge.
A Manuel Lanzini double plus strikes from Mark Noble and the impressive Marko Arnautovic gave the visitors a comfortable victory that took them above Huddersfield, for whom Joe Lolley scored and new 14 million pound (NZ$26.5 million) signing Alex Pritchard came on as a substitute.
The win was Moyes' 200th in the Premier League, a landmark reached by only three other managers.
At Selhurst Park, Crystal Palace's revival under Roy Hodgson continued with a hard-fought 1-0 victory over Burnley which moved them five points above the relegation zone.
Alan Pardew was another manager breathing easier after West Bromwich Albion beat Brighton & Hove Albion 2-0 to record their first win in 21 league games and his first in nine as their manager.
Both Albion goals came via headers from Chris Brunt corners, with Jonny Evans and Craig Dawson rising above Brighton's defence.