The Premier League leaders hit the front through Bernardo Silva's 16th-minute opener but an XI featuring five changes from Tuesday's 2-2 Champions Leaguedraw at Lyon did not have things their own way.
Callum Wilson found a deserved equaliser but Bournemouth were sunk by a familiar foe.
Sterling's 57th-minute goal was his eighth in six top-flight appearances against Eddie Howe's side and Ilkay Gundogan's simple finish 10 minutes from time meant an uneven outing ended with City extending their lead at the summit to five points.
Bournemouth were bright initially, with Wilson unable to turn home Ryan Fraser's low cross from a clever short corner, but City soon established control.
Gabriel Jesus, starting in front of Sergio Aguero, stretched to prod wide from Oleksandr Zinchenko's delivery and the Ukraine international's ball over the top of the Cherries defence brought out Asmir Begovic to thwart Leroy Sane. Silva was on hand to crash home the loose ball.
City goalkeeper Ederson pushed to safety when Wilson failed to get a decisive touch on Josh King's inswinging cross, but he could not keep out the Englandstriker's powerful 44th-minute header.
Zinchenko was forced off with badly damaged nose, having inadvertently clashed with his captain Fernandinho, while Sterling sought to get City going through a mazy dribble that ended with his deflected shot coming back off the upright.
The England winger's ninth goal of the season owed more to opportunism than ingenuity, however, as Begovic clumsily spilt Danilo's shot for him to find the roof of the net.
Fernandinho slid a left-footed shot narrowly wide, meaning Ederson's scrambled near-post save from Wilson came with nerves jangling.
But Sane engineered space down the left to tee up compatriot Gundogan, allowing Pep Guardiola to breath easily.
City given a warning for the second time in a week
Guardiola cautioned his players that their dominant record at home to Bournemouth – the three previous Premier League editions of this fixture returned an aggregate of 13-1 to City – counted for little but, as in the draw at Lyon, they showed defensive fallibility. They nevertheless continue to set a relentless pace in their title defence, with Liverpool needing a win in Sunday's Merseyside derby against Everton to remain within touching distance.
Wilson shows his class
Aymeric Laporte has a claim to being the Premier League's outstanding defender this season but the Frenchman and his centre-back partner Nicolas Otamendi endured a thorough examination from the excellent Wilson. Recently capped by England, the striker's pace, power and intelligent movement caught the eye alongside his seventh league goal of the season – one fewer than the division's joint-top scorer Sterling.
