Hardly for the first time, Heung-Min Son delivered yet another outstanding performance while keeping up a goal record which is none too shabby, either.
Spurs snatched their deserved lead four minutes later, when Son tapped in for his eighth goal of the season.
There are not many thing Kane has failed to do for Spurs in terms of scoring - but one thing is score in five successive Premier League home games.
The forward went into game on 96 goals, one behind Teddy Sheringham, but has now moved ahead of the legendary striker on the list with 98 goals.
Kane has now scored 20 league goals this season, three more than the next most prolific, Liverpool's Mohamed Salah.
He did have a rare off-day against West Ham in Spurs' last Premier League outing though, with the Hammers digging deep to get a 1-1 draw.
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Everton handed a debut to new £27m man Cenk Tosun following his move from Besiktas, but it was the home side who were the first to threaten after 15 minutes when Son flicked Eriksen's wicked free kick narrowly over the crossbar with Jordan Pickford rooted to the spot.
The Turkish global made a number of positive runs against Tottenham in the first half and if his support was better, he might have capped his debut with a dream goal at Wembley.
Everton tried to find a way back into the match and, with five minutes remaining until the break, Tosun got down the left but produced a disappointing cutback with colleagues well placed.
However, if Allardyce can sort out the problems behind him, Tosun could have a bright future with the Toffees.
Tottenham are unbeaten in their last 10 Premier League games against Everton (W5 D5), their longest such run against them since a run of 13 between 1997 and 2003.
They weren't holding back and kept attacking poor Pickford.
Not even the prolific Kane has managed that feat, but the England global 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.