City began the Premier League season with a win at newly promoted Brighton, before being held by Everton in their second game.
After falling behind to an excellent Charlie Daniels strike in the 13th minute, City created the better opportunities.
And they were level on 21 minutes when Gabriel Jesus got on the end of David Silva's precise through ball to beat Asmir Begovic.
The Brazilian striker almost added a second from Benjamin Mendy's cross but he couldn't direct his effort on target.
But deep into added time Sterling struck from close range to grab what seemed an unlikely victory.
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