Harry Kane has soared and scored before in the north London derby. The Tottenham striker rates his late headed winner against Arsenal in February 2015 as the finest goal of his career. He was at it again here in a moment to sum up the difference between bullying Spurs and fragile Arsenal.

Ben Davies’s cross from the left was inviting and Kane simply wanted it more than anybody in the vicinity. Timing his leap to perfection, he appeared to hang in the air, towering above Laurent Koscielny, before planting the header into the far corner. Arsenal’s players shouted for a push on Koscielny and so did Arsène Wenger but the appeals smacked of desperation. It was Kane’s seventh goal in seven Premier League matches against Arsenal.

This was a demonstration of Tottenham’s superior power and fitness, as much as their technical ability. After a tepid first half, Kane’s goal was the prompt for them to put the boot on their rivals’ throats. They pressed down hard and as Arsenal lost their shape it was a wonder how it did not turn into a beating. The visitors owed a debt to Petr Cech for keeping the scoreline respectable.

Mousa Dembélé is fast becoming the ultimate midfield controller | Barney Ronay Read more

The structure belonged to Tottenham whereas Wenger flung on attacking players in an increasingly frantic attempt to find a solution. It almost came off. Alexandre Lacazette had lifted one shot high when well-placed and in stoppage time he ran away from Davinson Sánchez, who had misread a through-ball from another substitute, Alex Iwobi. Lacazette had only Hugo Lloris to beat but his low shot went past the far post. Had he scored, it would have got Arsenal out of jail. It would not have masked the gulf between the teams.

Arsenal’s optimism had surged after last Saturday’s 5-1 home thrashing of Everton when their new attacking signings, Pierre-Emerick Aubameyang and Henrikh Mkhitaryan, had starred, although Aaron Ramsey, scorer of a hat-trick in that game, was missing through injury. They believed they had turned a corner. They ran into a wall here. Where Kane and his team-mates made things happen, those in red shirts shrank. Aubameyang and Mkhitaryan were anonymous.

Arsène Wenger: 'It was more than just a derby' – video

How Arsenal needed a statement victory over a big-six rival away from home. They must go back to January 2015 to find one – 17 matches ago – the 2-0 success at Manchester City. How they needed an away win of any description. Their form on their travels in the league this season has been unacceptable and, as Wenger pointed out, it has not only been because of defensive lapses. They have struggled to score. They have managed 15 in 14 games.

Arsenal’s hopes of a top-four finish have been left to hang by a thread and Wenger was not about to sugar-coat the picture. They are six points off the pace and the gap could grow when Liverpool and Chelsea play their games in hand. “We could not afford to lose today,” he said. “It makes it much more difficult now [to finish in the top four]. There was more at stake than just the derby.”

Arsenal had a couple of flickers in the early running when they almost got through on counterattacks only to miss the final pass. Unusually, Mesut Özil was culpable but Wenger’s claim that his team ought to have had the game sown up because of those moments was overblown. They were also denied by an offside flag after Jack Wilshere put Aubameyang clean through on 13 minutes. The decision was fiendishly tight.

Tottenham had more of the ball in the first half but the chances were sparse. A big one came on 27 minutes when Christian Eriksen unfurled a wonderful delivery for Kane but he could not direct the header. Cech had pushed away from his own player, Shkodran Mustafi, early on while Eric Dier almost turned in Mousa Dembélé’s mis-hit.

Facebook Twitter Pinterest Harry Kane soars to score the winner. Photograph: Robbie Jay Barratt – AMA/Getty Images

The intensity had been strangely absent in the first period. Tottenham brought it with a vengeance after the interval and Arsenal cracked. Kane’s goal was another classic and Wenger had the good grace to admit his initial complaints were misplaced. “It is a regular goal,” he said. “We can only look at ourselves.” For the umpteenth time, Mauricio Pochettino called Kane “one of the best in the world”. So did Wenger. “He is a super striker,” the Arsenal manager said. “He scores against everybody.”

Tottenham attacked in waves. Kane put a free header wide from Dier’s cross before working Cech with a stinging volley while the goalkeeper pawed away Eriksen’s free-kick. Wilshere extended Lloris with a thumping shot but it was otherwise all Tottenham. Dele Alli and the substitute, Érik Lamela, could not finish while Cech kept out a Kieran Trippier volley. Lacazette almost salvaged a point but it was not Arsenal’s day.