The striker became the 17th player to score 100 goals for Spurs when what looked like a cross from the right flew over Jordan Pickford's head and into the far corner on 28 minutes.
It was one-way traffic after Kane's opener.
Christian Eriksen soon made it 2-0 after Pickford parried out Ben Davies' shot and it took less than a minute into the second half for Kane to make it 101 goals and 3-0, angling home as Davies crossed from the left.
We poured forward from there and might have added more goals but for Pickford's saves from Kane and Eriksen and Dele curled just wide late on. Everton rarely threatened.
Davinson Sanchez made a fine full debut, a clean sheet, three points - the perfect afternoon on Merseyside!
Mauricio Pochettino made two changes to the team that started against Burnley on August 27. Sanchez made his full debut having come off the bench late against Burnley, making it an all ex-Ajax back three, while Moussa Sissoko came in for Mousa Dembele. Meanwhile, Serge Aurier and Fernando Llorente were on the bench for the first time.
CHANCES: FIRST REAL OPENINGS FOR US - 16mins
After a relatively even start we produced a great move with Eric Dier playing wide to Davies, his cross picked out Sissoko whose shot from 12 yards was blocked, play continued and Davies' cross to the near post was flicked wide by Kane from a difficult angle.
GOAL: EVERTON 0-1 SPURS - HARRY KANE - 28mins
Kane brought up his century of Spurs goals to open the scoring and open his own account for the season on 28 minutes. Eriksen's corner came out to the striker on the right side, he whipped the ball back in and it drifted over Pickford into the far corner.
CHANCES: CLOSE AT BOTH ENDS - 29mins
We had a chance to quickly extend our lead when Kane got away down the right and his low cross looked for Sissoko bursting into the box, but the ball somehow evaded the ex-Newcastle man as well as Pickford and went out for a goal-kick. Moments later, Sandro Ramirez let rip from 20 yards for Everton but Toby Alderweireld made a great block to deny him.
Above: Christian Eriksen makes it 2-0.
GOAL: EVERTON 0-2 SPURS - CHRISTIAN ERIKSEN - 42mins
We extended our lead three minutes before the break to really deal Everton a blow. Dele Alli advanced down the right, his cross found Davies whose shot was palmed out by Pickford, but Eriksen was on hand to ram home the rebound. Dele spurned a good opportunity just before the break while Kane also drilled wide as we continued causing the Toffees problems.
GOAL: EVERTON 0-3 SPURS - HARRY KANE - 46mins
Whatever was said in the home dressing room at half-time seemingly went out the window as Kane doubled his tally within a minute of the restart. Davies found space on the left side and he sent in a low cross for Kane to steer home from eight yards.
SAVES: PICKFORD CALLED INTO ACTION - 52mins
We were in total control and playing some stylish football and it needed Pickford, the former Sunderland stopper and England starlet, to prevent us from adding a fourth goal. First, on 52 minutes, he made a flying save to keep out a spectacular volley from Eriksen then, within a minute, he was there again to deny Kane after Dele had got in down the right.
MAURICIO POCHETTINO SAID
"It was a very solid performance and I’m very pleased. It was an amazing effort from all the team. I’m very happy. It’s three very important points for us at the start of a busy period. I’m so pleased and I can only congratulate our players. It’s true we’re early in the season but it’s always important to win games and a win in this way really builds confidence. There are a lot of things we need to settle in the team, new players, build a different dynamic but we’re so happy in the way we played today."
"It was a tough game at the beginning until Harry crossed the ball and it went in! I think we got the confidence and the possibility to play football afterwards. We played a very strong game, we didn't give them much and we had our chances."
Everton (4-2-3-1): Pickford, Martina, Keane, Williams, Baines (c), Schneiderlin, Gueye (Vlasic 80), Sigurdsson, Rooney, Klaasen (Davies 46), Sandro (Calvert-Lewin 46). Substitutes (not used): Stekelenburg, Jagielka, Mirallas, Holgate.
Spurs (5-1-3-1): Lloris (c), Trippier, Alderweireld, Sanchez, Vertonghen, Davies, Dier, Sissoko (Winks 77), Eriksen (Dembele 70), Dele, Kane (Son 85). Substitutes (not used): Vorm, Aurier, Walker-Peters, Llorente.
Goals: Spurs - Kane 28, 46, Eriksen 42.
Yellow cards: Everton - Williams 5, Gueye 57, Rooney 88; Spurs - Alderweireld 13.
Referee: Graham Scott.
MAN OF THE MATCH
Harry Kane was named Man of the Match by our supporters on social media with 42 per cent of the votes.
Our assault on this season's UEFA Champions League begins with a glamorous Group H opener against Borussia Dortmund at Wembley Stadium, connected by EE, on Wednesday night. We were beaten heavily on aggregate by the Bundesliga giants in the 2015-16 Europa League round of 16, but will be targeting a much better outcome this time as we look to get some early points on the board. Kick-off is at 7.45pm.