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.
Heung-min Son opened the scoring for Spurs, tapping in from close range to give his side the lead at half-time, his fifth consecutive home league game of scoring.
Kane - from an unorthodox, somewhat fortuitous finish, added his second, this time from a perfectly whipped in Eric Dier cross.
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Christian Eriksen capped off the win for Spurs, which sees them move joint fourth in the league, while Everton are now five league games without a win.
Kane has now scored 20 league goals this season, three more than the next most prolific, Liverpool's Mohamed Salah.
Allardyce handed their recent £27million Cenk Tosun his debut at Wembley, who he admitted still has a lot to learn about English football, but will hope the Turkish worldwide will find the shooting boots that drew Everton to him In the first place. Everton will need similar to get anything here.
However, if Allardyce can sort out the problems behind him, Tosun could have a bright future with the Toffees.
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Dembele was also impressive on the ball, completing 97% of his passes against Everton, misplacing just two of his 66 attempts. "Luckily it went my way".
It looked as if Doucoure had inadvertently turned the ball in with his hand as he stooped for a header but the goal stood and denied Southampton a first league win in 10 games.
Even with that caveat in mind, few teams in the world can keep up with Spurs when they're in the kind of effervescent mood we witnessed at Wembley today.
Not even the prolific Kane has managed that feat, but the England worldwide did add another record to his growing list of accolades just two minutes into the second half when he netted his 97th Premier League goal for Tottenham, drawing level with Sheringham.
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However the name that appeared at the top of the clubs wanted list was Espanol manager Quique Sánchez Flores. I am absolutely happy with my situation, I do not have to explain anything else.