Kane added his second just before the hour mark with the visitors' defence embarrassingly flat-footed while Christian Eriksen finished one of the best moves of the match by sweeping home inside the last 10 minutes.
But Spurs, and once again Kane, now had the bit between their teeth as he took the clubs Premier League record for himself after arriving unmarked to meet Eric Dier's pinpoint centre with an untidy, but nonetheless equally effective finish.
The in-form England worldwide, who took his tally in all competitions this season to 28 goals, matched Teddy Sheringham's feat after he tapped in Son Heung-min's through ball after the restart before prodding in Eric Dier's cross on the hour mark.
Only Stoke City and West Bromwich Albion (both six) have won fewer points away from home in the Premier League than Everton (eight) this season.
Meanwhile, Everton legend Kevin Ratcliffe has commented on Wayne Rooney's recent performances, and wrote in The Liverpool Echo: "My big concern about Wayne is that when he comes out after half-time he looks a bit stiff, a bit leggy and I'm not whether that's all the games catching up with him".
The home sides managed to round off the scoring in the 81st minute as Dele Alli's delightful flick teed up Eriksen, who calmly slotted into the bottom corner.
And Son was at the heart of the move that put Tottenham two goals to the good just two minutes into the second half.
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The England striker tapped home two second-half efforts to surpass Teddy Sheringham's club record that had stood at 97, and take his tally to 20 league goals for the fourth season in a row. Son cuts in from the right and thunders a superb strike against the post.
They sit three points adrift of fourth-placed Liverpool in the table.
By next summer, Eriksen will have two years left on the £70,000-a-week contract that he signed at the start of last season, and Tottenham are keen to reward the Denmark midfielder, who has five goals and six assists in the league this season, with a better deal and stave off interest from clubs overseas.
"This way he's going to break all the records in the Premier League and in England, and it's fantastic because he helps the team every week".
TOTTENHAM: Lloris 6, Aurier 8, Sanchez 7, Vertonghen 7, Davies 7, Dier 7, Dembele 7, Eriksen 8, Son 9, Alli 8, Kane 8.
Indeed, the Toffees did not register a single shot on target this evening and have now lost their opening three games of a calendar year for the first time since 1960.