The hosts made two changes to the side who won 1-0 at Newcastle last time out, with Christian Benteke and Scott Dann replacing Michy Batshuayi and the injured James Tomkins.
The Citizens made four alterations from their Champions League loss at Tottenham, with Kevin De Bruyne, Leroy Sane, Vincent Kompany and Benjamin Mendy all handed starts.
The opening minutes saw both teams struggling to find any rhythm in their play, before Sterling passed up a clear chance with 11 minutes on the clock, failing to hit the target after being superbly set up by David Silva.
However, he made amends three minutes later as he raced on to an exquisite throughball from De Bruyne and lifted a fine right-footed finish over advancing goalkeeper Vicente Guaita to give City the lead.
The visitors continued to dominate the play after moving ahead, with Guaita having to make a solid block at his near post to deny the influential Sane, before getting in the way of a De Bruyne effort following a goalmouth scramble.
Mendy smacked a 25-yarder straight at the waiting Guaita to end a first 45 minutes which saw the visitors in control.
Early in the second half Guaita reacted smartly to grab a deflection off Dann which was heading goalwards, following yet another fluent Citizens build-up.
Just before the hour mark Sterling played a dangerous cross in from the right, but there was no-one on hand to convert for the visitors.
Within five minutes Sterling had doubled the lead himself, as he was in the right place to steer home a Sane cross-cum-shot flashed across from the left-hand side of the area.
Ten minutes from time the Eagles earned themselves a lifeline as James McArthur was fouled just outside the area and Luka Milivojevic made no mistake from the resulting free-kick, leaving Ederson well beaten with a shot into the bottom corner of the net.
However, there was to be no way back for the hosts when, in the final minute of normal time, De Bruyne powered forward and fed substitute Gabriel Jesus, who calmly buried his finish inside Guaita's near post.