Manchester City kept up the pressure on Liverpool with a hard-fought victory over dogged opponents that was not quite as straightforward as the final scoreline might suggest. Everton have still not managed to come from behind to win a game under Marco Silva though a much improved performance almost earned them a point against the defending champions.

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A draw would have done their neighbours a favour by opening up a seven-point gap at the top of the table, and Everton were on course until Riyad Mahrez struck from a set piece with 20 minutes remaining. That was slightly ironic since Everton’s chief second-half threat had come from Gylfi Sigurdsson’s delivery of free-kicks and corners, with Yerry Mina in particular using his height to advantage, yet on the balance of chances created from open play the visitors deserved the victory even before Raheem Sterling made sure of it with a late left-foot volley.

“It was a good game, this is a difficult place to come,” Pep Guardiola said. “It is only September but we still needed to win after Liverpool won. The season is long and we just have to keep going.” Marco Silva said he was encouraged by everything except the result. “There were a lot of positives, and in the second half we had a couple of big chances to affect the result, but we didn’t take them. City were more clinical with theirs.”

There was no goal after 53 seconds, as City had managed against Watford last week, but if anything the game got off to an even more eventful start. First Theo Walcott departed on a stretcher after being knocked out by taking the ball in the face, six minutes of treatment on the pitch having failed to bring him round. Walcott was taken to hospital before later being released. Then Ilkay Gündogan hit the bar while practically standing on the Everton goalline, Mahrez’s fierce cross reaching him too quickly to adjust his feet in time, and when Jordan Pickford made his second save of the afternoon from the industrious Mahrez barely five minutes of play had taken place.

Once the contest settled down a little a well-worked Everton move ended with Sigurdsson seeing a shot blocked by Fernandinho, before City took the lead midway through the first half. Again Mahrez was involved, breaking down the right to set up Kevin De Bruyne for a cross, predictably sent early and accurately into the space right in front of Pickford where Gabriel Jesus twisted his body impressively to score with a close-range header.

Sigurdsson brought a save from Ederson as Everton attempted to hit back but Mahrez tested Pickford again at the other end, putting his shot too close to the goalkeeper after bursting into space. When Everton did equalise after half an hour it was from an unexpected source, Séamus Coleman delicately lifting a shot over Nicolás Otamendi and Ederson from a narrow angle after the ball had broken free from Alex Iwobi. It was a shot, not a cross, and though Dominic Calvert-Lewin got a final touch on the line to claim the goal, the ball was going in anyway.

City should have gone back in front just before the interval when a fine pass from Mahrez found Jesus in the clear but brought only a hesitant finish, the striker taking a touch to switch feet and ending up missing the target. There were chances for Gündogan and De Bruyne at the start of the second half and a decent one for Richarlison too, who headed wastefully from a Sigurdsson free-kick. The Brazilian sensibly left the next set-piece opportunity to Michael Keane and Yerry Mina, who combined to bring a first-class save from Ederson, clawing the ball out from under his crossbar.

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City produced a stunning opening on the hour when an exquisite through ball from Mahrez played Sterling in behind a square Everton defence, yet from close to the penalty spot with only Pickford to beat the winger slid his first-time shot the wrong side of an upright. The travelling support took up a chant of “Sergio, Sergio” almost immediately, and five minutes later the Argentinian was introduced, though it was Mahrez, whose performance deserved a goal, who made the breakthrough. City had a range of options when De Bruyne was fouled by Mina on the edge of the area, but Mahrez took the free-kick and cleared the wall with a shot that beat Pickford’s somewhat unconvincing dive.

City were indebted to Ederson for keeping out Calvert-Lewin after Sigurdsson carved out a chance, though the home side’s search for an equaliser inevitably left gaps at the rear.

After Mahrez had accelerated into one Sergio Agüero arrived at his shoulder to take over and drive across goal, where Sterling was able to fasten on to a rebound and steer the ball over the line. The goal had to be confirmed by technology, but the City fans in the Bullens Road stand had the perfect view and were already celebrating.