This was far from a straightforward win for City, with the visitors in the game until the latter stages having pegged back the champions who had initially led through Bernardo Silva's strike.
Bournemouth started confidently and could have gone ahead after just four minutes as Ryan Fraser got in behind the Blues’ defence, but his cross was slightly behind Callum Wilson whose attempted back flick went wide.
The next meaningful action came on 12 minutes when Oleksandr Zinchenko, casually moving forward, whipped in a superb cross that Gabriel Jesus was inches away from converting.
City’s next chance resulted in a goal and Zinchenko was again the creator.
The Ukrainian spotted Leroy Sane’s run and dinked a lovely ball into the German’s path – but the onrushing Asmir Begovic managed to beat away his shot only for Bernardo to fire powerfully into the unguarded net.
Bernardo went close to adding another 60 seconds later, but his half-volley was cleared by the Bournemouth defence.
While the game was still only 1-0, so the visitors were still very much in the hunt – and that meant there was no drastic need to change their largely defensive formation which was built to contain the champions.
Bournemouth looked increasingly dangerous on the break and Josh King whipped in a cross that Wilson was inches away from converting, with Ederson doing well to push clear a curling cross.
Though City were dominating possession, the Cherries grew in confidence and there were a couple more nervy moments in the home defence with a couple of hurried clearances as half-time approached.
The warning signs had been posted.
It wasn’t a stellar City display by any means up to that stage and the slender advantage was finally snuffed out a minute before the break, as Simon Francis - allowed time and space on the right - clipped the ball in for Wilson to rise higher than anyone else and plant a header past Ederson.
By the Blues' own lofty standards, things needed to shift up a gear after the break, but it was the Cherries who threatened first through a Wilson and Josh King combination that ended with across that just evaded Charlie Daniels at the far post.
City came close to regaining the lead on 53 minutes thanks to a breath-taking run by Raheem Sterling.
The England winger danced by one, two, three, four, five and then six challenges before seeing his shot deflected onto the post by Steve Francis with the Etihad roof about to lift off.
But the Blues' supporters - and Sterling - didn't have to wait much longer for the net to burst.
City were much sharper, had much more energy and looked determined and focused to retake control of the game and on 56 minutes, Danilo fired a low drive that Begovic could only parry and - who else? - Sterling was on hand to thump the ball into the roof of the net.
In scoring (as he always does against the Cherries), he became the first player to score in six successive games against the same opponent - it was also his ninth of the campaign.
The visitors were still in the hunt and again, looked threatening on the counter-attack, but a difficult encounter was finally put to bed 10 minutes from the end when a neat passing move ended with Sane's low cross being turned home by Gundogan from close range.
Sane went agonisingly close to adding a fourth on 84 as he tormented his marker before seeing a powerful drive superbly deflected over by Begovic.
So, unbeaten in 20 games, five clear (for 24 hours at least) and our 100% home record remains intact - but Bournemouth didn't make it easy!
Star man:Leroy Sane
City's German winger never stopped working and made one goal on an industrious afternoon.
What it means…
City extend the lead at the top of the Premier League to five points ahead of Sunday's fixtures.
On Sergio’s absence and resting David Silva::
“There are a lot of games and we don’t want to take risks of injuries. One month or a month-and-a-half loss with nine or ten games is too much.
“I don’t know if Aguero will be fit for Watford. We will see.
“You ask me why Sergio and David Silva don’t play, and I’ll give you an argument for each one.
“I base my decisions on doctors, physios, instinct and what I see. There are many reasons.”
On beating Bournemouth:
“It was a good win. After the Champions League, it is always tough.
“People think it’s easy for us but that is not the truth, so I like to win in this way.
“We had a lack of rhythm in the first half. Bournemouth are so fast up front and so hard to play against. They put so many balls into the box – they are so strong.
“We were flat but after, we were more aggressive, and we changed the game. I asked them for more at half-time, but my speech was ridiculous because in the first five or ten minutes, it didn’t work – it was the same as the first half!
“Raheem Sterling changed the game and Leroy Sane was fantastic too. I don’t know what he ate but it will be on the menu for the next games!
“What we have done in the Premier League is incredible and it means a lot to me.”
City travel to Watford on Tuesday evening for what looks like a tricky test against the Hornets.
Though the Blues have won handsomely on the last two visits, scoring 11 goals in total, this is likely to be a closer game, with the hosts making a solid start to the 2018/19 campaign.
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