Sterling had already missed a glorious chance before opening the scoring on 15 minutes as City rarely looked troubled despite having failed to beat Palace on their two previous meetings.
Another cool Sterling finish looked to have ended the game as a contest 27 minutes from time.However, Pep Guardiola’s men had to see out a nervous finale to ramp up the pressure on Liverpool when Luka Milivojevic’s free-kick gave Palace hope of a dramatic comeback.
Instead, City broke to seal the three points on the counter-attack when the outstanding Kevin de Bruyne teed up Gabriel Jesus to make it 3-1.
The trip to south London kicked off a massive week for City with Tottenham to visit the Etihad twice in the next six days for the second leg of their Champions League quarter-final and a vital Premier League clash with implications for both the title and top four race.
City must overturn a 1-0 first leg deficit against Spurs to progress to the Champions League semi-finals on Wednesday. Yet, Guardiola’s starting line-up seemed stronger than the one he named in defeat to Mauricio Pochettino’s men in midweek.
Captain Vincent Kompany, Benjamin Mendy, De Bruyne and Leroy Sane returned among four changes and City quickly settled into total control of the game.
Sterling should have opened the scoring on 11 minutes when after a patient build-up, David Silva got to the by-line and picked out the England international who somehow sliced wide with an open goal at his mercy from close range.
After a similar sitter at Burnley last season, Guardiola quickly hooked Sterling in his frustration.However, Sterling showed his maturity and why he is a contender for player of the year awards by redeeming himself in emphatic fashion just four minutes later.
A rare Palace foray forward left them exposed at the back and De Bruyne’s sensational through ball allowed Sterling to break clear before side-footing the ball high beyond Vicente Guaita.
Palace inflicted City’s only home league defeat of the season with a shock 3-2 victory in December thanks in part to a wonder strike from Andros Townsend.
But the hosts were camped in their own half and unable to spring a counter-attack for the pace of Townsend or Wilfried Zaha as City enjoyed over 75 percent possession in the first 45 minutes.
Guaita made a smart stop at his near post to prevent Sane killing the game as a contest after a slick solo run and the Spanish goalkeeper also played his part in a goalmouth scramble that Palace somehow escaped unharmed from in first-half stoppage time.
A similar pattern continued after the break and City finally struck the killer blow just after the hour mark when Sane’s driven cross was turned home at the back post by Sterling.
City still had to survive a nervy finale after Milivojevic’s free-kick from the edge of the box reduced the deficit to 2-1 nine minutes from time.
But they withstood an aerial bombardment and made the game safe when substitute Jesus took his time before beating Guaita at his near post.