If Everton are to break through the Premier League’s glass ceiling and move upwards into the top six at some point they are going to have to start beating the teams likely to finish above them. There was no sign of that here. Tottenham swatted away Everton’s challenge with almost contemptuous ease, leaving Ronald Koeman to reflect that even significant summer spending does not appear to have made the job any easier.

The Everton manager had been highly complimentary about Spurs beforehand, describing them as one of the strongest teams in the Premier League. That is hardly an exclusive: Tottenham have looked the part for several seasons now, but lest anyone underestimate Mauricio Pochettino’s players after a slowish opening this was an object lesson in how to control a game through application and intelligence.

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Davinson Sánchez, making a solid start as the central defender in a back three and looking as comfortable as if he had been there all his life, must be wondering if all the reports about the toughness of the Premier League are correct. Everton is supposed to be one of the more daunting venues to visit, after all, yet Spurs were so far ahead by the end of the first half they could practically afford to take the second half off. The home supporters marked Goodison’s 125th birthday with an uncharacteristic 45-minute silence. Long before the end of the game the Spurs fans were the only ones making any noise and the Gwladys Street end was half empty.

“We had problems coping with their diamond in midfield,” Koeman said. “Because of the movement of Alli and Eriksen they were always able to easily find their free player.”

With August out of the way Harry Kane was free to start scoring again, and his 100th goal for Spurs was one to remember. Jordan Pickford, at least, will not forget it in a hurry, Everton’s new goalkeeper being unaccustomed to being beaten by dipping shots from a long way outside the box, but intentionally or otherwise Kane put the ball in the one place where no one could reach it. That included his Spurs team-mates, too, so if Kane was attempting to cross after retrieving an overhit corner – he later said he had not intended a shot – it was a poor effort.

So in one sense Tottenham’s opener came from nothing, though by that point the visitors had weathered Everton’s early storm – if one chance for Sandro Ramírez from a glorious cross by Cuco Martina can be thus described – and begun to dictate play on their own terms. Spurs not only pressed better than Everton, constantly forcing the defence and midfield to turn the ball back towards their own goal, they passed better than the home side too. Christian Eriksen’s poise and distribution was particularly notable, completely overshadowing Gylfi Sigurdsson’s contribution on his home debut, and for a while it appeared that almost any Spurs player in any part of the pitch had the capacity to find Ben Davies overlapping in space on the left. “He’s a great player but today he was fantastic,” Pochettino said. “Everyone played well though, we haven’t had an easy period and that was the solid performance we needed.”

Moussa Sissoko could have put Spurs ahead from a Davies cross even before Kane broke the deadlock, but with the Everton defence scrambling he was unable to come up with a convincing finish. Eriksen had no such problem in extending the Spurs lead three minutes before the interval. Pickford had partly redeemed himself with a point-blank save from Ben Davies, but the goalkeeper could not hold the ball and Eriksen was on hand with an emphatic follow-up.

Everton fell apart a little after that. They had to negotiate three or four minutes to reach the sanctuary of the dressing room, yet in that short time they could easily have shipped another couple of goals. You would have put money on Dele Alli scoring when Kane picked him out in front of goal but the England player could not adjust his feet quickly enough, then with virtually the last kick of the first half Kane rolled a low shot just inches past a post. There was audible grumbling by this stage, and the half-time whistle was greeted by more than a few boos.

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Koeman’s response was to send on Tom Davies and Dominic Calvert-Lewin, yet remarkably Everton failed to shut down Ben Davies on the left. It took less than a minute after the restart for the wing-back to make them regret that, easily picking out Kane for the striker to register his 101st Spurs goal with a languid finish.

Whatever the Everton manager had said at the interval was now irrelevant; his side were involved in a damage limitation exercise. Namely, trying to keep the final score more respectable than the one Liverpool ended up with at Manchester City. Thanks to a flying save from Pickford that kept out an Eriksen volley and a couple of late misses from an anonymous Wayne Rooney, the score remained unchanged. Officially, at least, this was not as heavy a defeat as the one Liverpool suffered. While there might be one or two Everton fans who will take a hollow crumb of comfort from that knowledge, Liverpool were playing away from home. And they were down to 10 men.