Arsenal have made a series of statements this season, as Arsène Wenger plods the long distance towards the ultimate vindication, such as beating both of the Manchester clubs at the Emirates Stadium and even getting a point at Stoke City. Some things, however, never seem to change.

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Chelsea remain their nemesis, even without José Mourinho as their manager, winding them up and, well, beating them, and it was another character who they have come to loathe that was the match-winner here.

Diego Costa was involved in a red-card flashpoint – as he was in Chelsea’s 2-0 win over Arsenal in September – but this time, there could be no recriminations. The striker was simply too fast for Per Mertesacker in the 18th minute and, when the Arsenal defender slid in and got none of the ball, there was an inevitability about his dismissal for a last-man foul.

Costa promptly showed the clinical side to his game in front of goal, when he pounced to meet Branislav Ivanovic’s cross and steered a shot beyond Petr Cech. He is back in business after his toils during the Mourinho implosion, with a sixth goal in six matches under the interim manager, Guus Hiddink.

It set Chelsea on their way to a sixth win over Arsenal in nine meetings, excluding the Community Shield – the other three have been drawn – and Hiddink’s only criticism of his players could be that they failed to find the killer second goal. Cesc Fàbregas was outstanding, setting the tempo in midfield and impressing with his poise and vision while Willian was also dangerous.

Arsenal’s regrets centred upon the red card and Wenger blustered about the decision from the referee, Mark Clattenburg, as having been “quick and harsh”. Yet he was simply angry that his team had been forced to confront such streetwise opponents with 10 men for so long. It made for what he suggested were exceptional circumstances and deepened the sense of regret.

Wenger felt that Chelsea might have been there for the taking had his team retained 11 players but the visitors were comfortable even before Mertesacker’s dismissal, with Oscar and Willian working Cech. Arsenal could point to a miscued volley from Joel Campbell and a half-chance for Mathieu Flamini, which went begging, before Mertesacker stretched into the challenge that would shape the game.

He never looked like getting to the ball first, following Willian’s probing pass for Costa on the counterattack and, in what felt like the blink of an eye, the Chelsea striker had tumbled into a series of exaggerated rolls and everybody inside the stadium knew what was coming next.

Costa took five minutes to salt the wound. Wenger had made the decision to sacrifice Olivier Giroud for Gabriel Paulista – the replacement defender that he needed – and, if the Emirates crowd did not like the substitution, Giroud was even less happy. The striker gestured with his outstretched arms and initially dragged his heels as he made his way off, until Flamini came across to chivvy him along. Mertesacker and Giroud would be spotted in their club suits behind the Arsenal bench during the second half.

Gabriel versus Costa rekindled memories of the clash at Stamford Bridge earlier in the season, in which the former had been sent off for a little dig at the latter; Costa’s reaction that day had sealed the deal.

Gabriel was not up to the pace of the game when Ivanovic drove over a cross and how Costa made him pay. Tip-toeing in front of him, Costa touched home at the near post before pointing out the name on the back of his shirt to the Arsenal supporters.

Chelsea might have been further in front by the interval only for Cech to save from Costa at the near post and Nacho Monreal to clear off the line from Ivanovic’s header. And yet they could, equally, have been level. Following Aaron Ramsey’s smart dink over the top, Flamini – all alone 10 yards out – flew at the chance like Hong Kong Phooey out of a filing cabinet. To loud groans, he lifted his flying kick over the crossbar.

Fàbregas has been referred to as a rat and a snake during what has been a trying season and his every touch was booed by the fans of his former club. But he responded in fine style and he might have had a penalty on 56 minutes when Laurent Koscielny checked him inside the area. Clattenburg was unmoved.

Costa played the pantomime villain to the last. He had gone down in need of treatment for a slight knee injury, to the scorn of the home support, when his number went up and Loïc Rémy prepared to replace him. Costa had the treatment, got up and walked off at low speed. He would raise his hands to applaud the Arsenal crowd before strolling away down the tunnel.

Wenger played his trump card with the introduction of Alexis Sánchez on 57 minutes for his first football since 29 November and a hamstring tear, and the forward made a difference. But it was not enough of one. He had Arsenal’s one flicker of a chance in the second half following a scramble only to swing and miss his kick. The full-time whistle brought several Arsenal players to their knees. They know all about losing to Chelsea but this one cut them to the core.

Man of the match Cesc Fàbregas (Chelsea)