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The irrepressible Son Heung-Min and Christian Eriksen were Spurs’ other scorers in a win which extended Tottenham’s unbeaten run against Everton to a remarkable 11 matches.
Judging by the gulf class in this game, that run may go on for a while yet, with Tottenham now level on points with Everton’s fourth-placed local rivals Liverpool.
Kane now has 98 Premier League goals for Spurs compared to England predecessor Sheringham’s 97 for the north London club.
But he is still well short of Spurs legend Jimmy Greaves’s tally of 220 goals in 321 appearances in the old First Division.
Everton manager Sam Allardyce gave a debut to Turkey striker Cenk Tosun. The 26-year-old was signed from Besiktas earlier this month, forgoing Champions League football in Istanbul for a new start on Merseyside.
Tosun’s first touch was on a leg, not the ball as he caught Davinson Sanchez in the ninth minute.
By then, Tottenham had weaved their familiar patterns through and around the Everton midfield without creating a clear opportunity in front of goal.
The closest they came was when Kane slipped a pass in front of Dele Alli, only for Mason Holgate to avert the danger with a composed challenge on his fellow 21-year-old.
That was until Eriksen curled in a free-kick and Son flicked a back-header just wide.
Everton finally roused themselves with Wayne Rooney dragging a left-foot shot just wide before he had a goal disallowed for offside after 22 minutes.
Tosun rose highest at a corner, guided his header goalwards only for Rooney to make his move fractionally too soon before nodding home.
At the other end, Kane forced three saves from Jordan Pickford in as many minutes, before Tottenham took the lead in the 26th minute.
Serge Aurier drifted forward unmarked and was found with a sweeping Eriksen pass. The wing-back sprinted into the area before dragging his shot into the path of Son who gleefully steered it home.
– Irresistible –
If the rest of the first half fell flat, the second period burst into life with Son and Kane proving an irresistible combination for Everton to contain.
Son fooled Jonjoe Kenny with a delightful turn on halfway. He then sprinted into the Everton area and slipped a pass into Kane’s path for an easy tap-in.
Son then fed Alli only to see the England international’s shot slice into the side netting.
South Korea’s Son was not finished there, however, and drove a rasping shot from the edge of the area against the base of the post.
A third Tottenham goal was inevitable. It duly arrived in the 59th minute when Eric Dier curled in a diagonal cross and Kane skipped his shot over Pickford.
If that was a curious finish, it nonetheless took him past Sheringham’s mark during the much-travelled striker’s time at White Hart Lane.
Still Tottenham poured forward, knowing a five-goal victory would take them above Liverpool and into the Champions League places –- at least for a day, with the Reds facing runaway league leaders Manchester City at Anfield on Sunday.
The overworked Pickford could do nothing when Son again combined with Alli who back-heeled the ball into Eriksen’s path nine minutes from time for the Dane to sweep home his side’s fourth goal and set the seal on a comprehensive victory.
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