Coming away with nothing again had seemed inconceivable after an opening 45 minutes which Arsenal dominated, taking the lead on five minutes when Theo Walcott seized on Alexis Sanchez's weighted pass to round off a thrilling three-man move.
City almost equalised immediately when Sterling headed just wide but they looked strangely hesitant going forward and could have fallen further behind as Arsenal played with confidence and menace.
City came out for the second half completely transformed, hitting back almost immediately as David Silva found Sane with a precision pass that allowed the German to score his first goal since his 46-million-pound ($57.49-million) move from Schalke.
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Arsenal wilted alarmingly and frequently came off second best in the physical contest. It was no real surprise when City went further ahead after Kevin de Bruyne supplied Sterling with a raking crossfield pass on 71 minutes.
Sterling cut inside on his left foot and beat Petr Cech at his near post with a low shot. It was the first time since 2012 that City had won at the Etihad after trailing at the interval.
"It was good to get on the scoresheet," said Sterling. "We stayed in the game and got the winner."
The result dropped Arsenal to fourth place.
(Reporting by Neil Robinson; Editing by Clare Fallon)
