Sterling's joy was short-lived though as he was shown a second yellow card, resulting in a red, for going into the crowd to celebrate his goal -- his second in a week.
A Charlie Daniels screamer gave Bournemouth the lead after 13 minutes but City were level eight minutes later when Gabriel Jesus, preferred to Sergio Aguero, poked home.
Both sides struck the woodwork after the break -- Josh King for Bournemouth and Nicolas Otamendi for City -- and Bournemouth seemed set for their first point of the season.
But seven minutes into stoppage time Sterling linked up with Danilo in the area and his weakly-hit shot took a deflection on its way past a helpless Asmir Begovic.
Sterling, who scored City's leveller in a 1-1 draw against Everton on Monday, ran to the City fans who spilled over barriers with police moving in to restore order.
(Reporting by Martyn Herman; Editing by Clare Fallon)
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