The Brazilian striker pounced when Nick Pope failed to deal with Bernardo Silva's strike in the third minute.
And Sterling doubled the advantage just before the break when finishing Ilkay Gundogen's cross from close range.
The visitors controlled the game throughout and finished with 75% possession while adding 19 attempts on goal.
Here are the RATINGS.
1. Nick Pope 6
Will be disappointed to have dealt with Silva's early effort, but continues to heighten his profile as an excellent shot-stopper. Denied Sterling with his legs after the break and made a save at full stretch to deny Mahrez.
2. Matt Lowton 5
Failed to negate the threat of Sterling in the build up to City's opener. Had the opportunity to cut out the supply, but unconvincing in the challenge. Got stuck in after the break, but troubled by City's width.
3. Erik Pieters 5
Lost Jesus for the away side's opener when Pope spilled Silva's strike. Mahrez always able to find space in-behind and the full back struggled to contain the Algerian's trickery.
4. James Tarkowski 6
Pulled up for his positioning when caught on the wrong side of Silva from Sterling's pass in the build up to City's first. Otherwise showed great strength on and off the ball, ushered City's forwards into safe areas and made a magnificent block to deny Mahrez.