The clock is ticking down on White Hart Lane and Tottenham Hotspur appear determined to wring every last drop of satisfaction from the old place. As things stand, the plan is to vacate their home since 1899 for the whole of next season before they return to a rebuilt 61,000-seat stadium in 2018-19. It meant the visit of Everton could be advertised as the sixth last league fixture at a ground that holds many dear memories.

It would go down as Tottenham’s ninth straight Premier League win here. Everton had brought the best record in the division during the calendar year but they were swatted aside by a combination of Harry Kane’s ruthlessness and Tottenham’s overall physicality.

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Mauricio Pochettino’s players are on such a roll that it feels as though visiting teams are on the back foot before they kick off. The Everton manager, Ronald Koeman, commented on how difficult it was to play against Pochettino’s 3-4-2-1 system and for the best part of 81 minutes Tottenham called the tune.

Romelu Lukaku’s late goal threatened an unlikely Everton comeback but Dele Alli’s lovely stoppage-time finish disabused them of that notion. Pochettino’s celebrations were frenzied and, although the Everton substitute Enner Valencia plundered an even later goal, it would count for nothing. Tottenham were full value for a result that keeps them – just about – in the rearview mirror of the league leaders, Chelsea.

Kane had entered the game with three hat-tricks since the turn of the year and he ought to have had another one. He was denied in two one-on-ones by the goalkeeper Joel Robles, with the second on 82 minutes a particularly clear opportunity. However, as it was, Kane scored twice and yet again he made the difference. The 23-year-old is now the outright top scorer in the league this season, with 19 goals – one more than Lukaku – and he has 24 in all competitions.

Facebook Twitter Pinterest Dele Alli scores Tottenham’s third goal in the Premier League match against Everton. Photograph: Ian Walton/Getty Images

It had been tight and tactical at the outset, with Everton closing the space between the lines, but Kane changed the complexion of the afternoon with the breakthrough. It came out of nothing but this is the sort of thing that can happen when he is on the field. Everton did not feel any danger when Kane picked up possession 40 yards out and Idrissa Gueye allowed him to turn and move towards goal. Gareth Barry was not worried, either, and he dropped off. It was not as if Kane had the ability to explode a right-footed shot past Robles from 25 yards. Oh, wait.

It was the prompt for a furious 10-minute period of Tottenham pressure, with Robles blocking from Kane; Christian Eriksen dragging wide after being sent through by Kyle Walker and Victor Wanyama side-footing the ball against a post from just inside the penalty area.

Tottenham won the right to play their football, with Mousa Dembélé and Wanyama coming to boss the middle of the pitch. The home team’s control was impressive. Koeman had started with a narrow 4-2-2-2 formation, with Barry in for the young forward Ademola Lookman and Ross Barkley pressed high alongside Lukaku. Koeman switched to 4-3-3 on 64 minutes, when he sent on James McCarthy and Kevin Mirallas but, by then, his team were two goals down.

Everton had tried to play out from the back when Robles rolled the ball to Morgan Schneiderlin only for Alli and Dembélé to swarm all over him. When Schneiderlin tried to pop the ball off to Ashley Williams, he found only Alli, who touched through for Kane. One on one with Robles, there was an inevitability about the outcome. Kane took one touch to set himself and another to shoot low into the net.

Tottenham had policed Lukaku diligently, with Toby Alderweireld and Jan Vertonghen – who know the striker from the Belgium national team – each making excellent tackles on him in the early running. Eric Dier, Tottenham’s third centre-half, also stretched into a saving slide challenge on Tom Davies in the 13th minute.

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However, like Kane, Lukaku needs only a sniff of a chance and he got it when the otherwise outstanding Vertonghen got his feet in a mess as he backpedalled following Mirallas’s ball forward and slipped over. Lukaku raced through to ram a low shot past Hugo Lloris and a finale that had scarcely been trailed was set up.

Alli had headed straight at Robles earlier in the second half while Vertonghen was denied by the goalkeeper after a storming run upfield and, when Kane fluffed his dinked finish when clean through – Robles saving well – Everton could dream of snatching a point.

Alli’s goal put the result beyond doubt and it followed a piece of quick thinking by the substitute Harry Winks. Rather than seek to run down the clock, he lofted a free-kick into the area for Alli, who touched a flick with the outside of his boot into the far corner. Valencia had the last word, when he stole in to convert from Séamus Coleman’s free-kick – after a Tottenham defensive breakdown. But Pochettino’s team had done more than enough.