After scuffing a great chance wide early on, Sterling made amends with a superb finish from Kevin De Bruyne's incisive through-ball on 15 minutes.
Vicente Guaita's fingertips kept Leroy Sane's arrowed shot out to keep it 1-0 at half-time but Sane's first-time cross was turned in by Sterling for his 17th goal of the season on 63 minutes.
Luka Milivojevic's 81st-minute free-kick gave the hosts hope but substitute Gabriel Jesus swept in De Bruyne's pass in the 90th minute to wrap up City's victory.
It was Man City's ninth consecutive league win and they are now two points behind leaders Liverpool, with a match in hand, after the Reds defeated Chelsea later on Sunday afternoon.
Crystal Palace remain in 13th on 39 points.