City’s victory maintains their 100% home record in the league this season, with eight wins out of eight, but this latest triumph was not as straightforward as the scoreline might suggest.
All seemed to be going to plan for Pep Guardiola’s side when Bernardo Silva smashed them ahead early on, with the Portuguese midfielder following up after Asmir Begovic had denied Leroy Sane.
But the home side, with David Silva on the bench and top scorer Sergio Aguero absent altogether after he complained of a muscle discomfort, relaxed a little too much before half-time and were deservedly punished.
To the displeasure of the home fans, the Cherries were given far too much time and space to build outside the box and Simon Francis swung in a perfect cross for Callum Wilson to rise and head home.
Bournemouth threatened again after half-time when King’s looping cross almost dropped in at the far post, while City looked surprisingly short of urgency or inspiration.
It was Sterling who supplied the spark they were missing, to continue his remarkable run of scoring against the Cherries.
The England forward followed up after Begovic could only parry Danilo’s low shot and Nathan Ake failed to clear, firing into the roof of the net for his 10th goal in seven games against the Cherries.
Bournemouth, looking to avoid a fourth successive defeat, did not lie down and some dangerous balls into the box ensured some nervous moments for the Etihad faithful.
It took Gundogan’s close-range finish to make sure of the points with 11 minutes remaining, with the impressive Sane providing the assist with a precise pull-back.