Everton lost winger Theo Walcott in the opening minutes after he was struck in the face by a Sterling cross and had to be taken to hospital for checks.
After Ilkay Gundogan had hit the bar for City, Pep Guardiola’s side grabbed a 24th-minute lead when Brazilian Jesus stooped at the back post to head in a Kevin De Bruyne cross.
A determined Everton got back on level terms, though, nine minutes later, when after poor clearance from Fernandinho, Seamus Coleman lofted the ball goalwards.
Dominic Calvert-Lewin then dived to make sure with a header on the line.
Sterling made sure of the three points, blasting home off the underside of the bar after a low cross from substitute Sergio Aguero was deflected out off the foot of Everton keeper Jordan Pickford.