In the seventh minute of added time, the Manchester City technical area exploded into life as Raheem Sterling wheeled away after earning three points on a picture-postcard afternoon at Bournemouth. His celebrations, which included jumping into a frenzy of travelling supporters who had spilled on to the pitch, earned him a second yellow card, as Pep Guardiola’s side took victory in the cruellest manner following a superb and often heated contest between two teams coached by perfectionists.

It was the second time this week that Sterling had come up with the goods, after his equaliser at Everton on Monday. City were heading for a second successive Premier League draw before he nicked the winner, firing home Danilo’s low cross with only seconds left. It was a bonkers finale to a game that concluded in the 101st minute. Gabriel Jesus had drilled in a first-half equaliser for City after the Bournemouth defender Charlie Daniels filed an early contender for goal of the season, with a rasping half-volley, that pinballed off the crossbar and in from 20 yards. By the final whistle, when Eddie Howe had his head in his hands, Daniels’ spectacular strike seemed a distant memory.

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City made life difficult, and Bournemouth must take spirit from how they stifled Guardiola’s side for so long, but his side extended their unbeaten record to 11 matches and broke hearts in the process. Howe and his players were aggrieved with the referee, Mike Dean, for penalising the substitute Lys Mousset in the buildup to City’s winner, although the hosts could count themselves lucky not to have finished the game with 10 players themselves. The numbers are not adding up at the moment for Bournemouth, pointless after three games.

Guardiola made four changes from Monday’s draw, with Sergio Agüero among those to drop to a bench brimming with more than £202m worth of talent. Benjamin Mendy and Bernardo Silva, both of whom arrived from Monaco over the summer, made their first starts while Guardiola shifted to a back four, with another summer signing, Danilo, at right-back. Howe handed Jermain Defoe his first Premier League start for Bournemouth, after he re-signed for the club over the summer.

It was a measure of how impressively they began that a first league goal this season, after 13 minutes, took nobody by surprise. After a swell of pressure Dan Gosling’s effort was headed away by Vincent Kompany, but only so far as an onrushing Daniels, who needed no second invitation to thrash the ball home. He rifled an unwavering half-volley towards goal, which smacked the inner frame of Ederson’s crossbar before crashing in past the City goalkeeper.

City looked shell-shocked and could have fell further behind but for Ederson, who denied Defoe from close range after the England striker lost his marker, Mendy, to meet Andrew Surman’s neat first-time cross.

Bournemouth had City rattled but not for much longer, with David Silva crafting a fine equaliser eight minutes later. The move was started and finished by Jesus, who took a quick free-kick before Silva slid him through with a beautifully weighted pass and he rolled the ball beyond Asmir Begovic.

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Sixty seconds later, Jesus again got the better of Nathan Aké, robbing him of the ball before he was fouled by the Bournemouth defender 25 yards from goal. Aké was arguably the last man and fortunate to escape with only a booking. Jesus then shinned an effort wide after a dangerous ball by Mendy. Bernardo Silva, too, had a golden chance but spurned what was virtually a free hit at Begovic’s goal. City had suddenly found their groove and dominated the rest of the first half.

After the interval, Steve Cook put in a bruising late challenge on Jesus but both came away from the tackle unscathed. City continued to pry for an opening as Kevin De Bruyne grew increasingly instrumental. It was his sweeping cross that eventually found its way to Bernardo Silva, but the forward got himself in a tangle at the back post. Then Bournemouth went close. King struck an upright with Ederson beaten all ends up.

City continued to pour forward and Guardiola called upon Agüero in search of a winner. He won a corner with which De Bruyne laid the ball on a plate for the unmarked Nicolas Otamendi – erratic at times in his own penalty area – whose tame header struck the post. Then came Leroy Sané, in place of Jesus.

Bournemouth had one more warning, only for Daniels to get his team-mate Aké off the hook. The Dutchman threw a stray boot at Mendy’s devious cross but Daniels slid in to deny an onrushing David Silva the simplest of finishes. But after Lee Probert signalled for five added minutes, City eventually found a shattering winner.