Sterling had already wasted a great chance by the time he shot high into the net to give City the lead after 15 minutes, and he wrapped up the points with a close range finish after the break.
Luka Milivojevic made it an unexpectedly nervy finale for Pep Guardiola's side when he whipped in a free kick from the edge of the box nine minutes from the end, but Jesus had the last word as he rounded off a 90th-minute counter-attack.
City went a point clear with the win, only to fall back to second once again following Liverpool's victory over Chelsea.
Sterling has 21 goals for City this season in all competitions and Guardiola praised his capability to shrug off incidents that don't go his way.
"The Burnley miss was worse than today," he told a news conference.
"We have to learn from golf and other sports like that -- the most important action is the next one. Sometimes you miss, you have to forget about it and go for the next one. He made an incredible finish and the pass from Kevin was outstanding. The finishing from Sterling was incredible."
Sterling said he was relaxed about the situation and added that City can't afford any slip-ups in their title race with Liverpool.
"When I was younger you always want to score your first opportunity and sometimes it doesn't happen like that," he said.
"You don't beat yourself up. There's 80 minutes left, you just have to be ready for the next opportunity. That's the mentality, just keep going and the chance will come."
City started purposefully, fashioning an early half-chance for David Silva, who was just unable to connect cleanly with a low Kyle Walker cross. They missed a golden chance after 10 minutes when Silva did brilliantly to reach the byline and cut the ball back to the unmarked Sterling, who could only roll it past the post.
But Sterling made amends on the quarter of an hour mark when Kevin De Bruyne's pass down the right enabled him to advance into the area and fire home an angled finish.
The home side soon had an injury concern, Jeffrey Schlupp limping off after a collision and being replaced by Cheikhou Kouyate, with City continuing to threaten and Scott Dann doing well to clear another Silva cross.
Sterling showed the other side of his game, tracking back to help Walker deal with Wilfried Zaha, and Aguero fired over as City maintained their control.
But Palace showed a flicker of attacking life soon afterwards, Milivojevic crossing and Christian Benteke heading down before City hacked clear.
Soon after the half-hour, Sane forced a good save from Vicente Guaita when he fired an angled drive towards the near post.
Guaita was in the thick of the action again when the ball fell to De Bruyne and he fired goalwards, and then Benjamin Mendy's strike was straight at the Palace keeper.
But the visitors doubled their advantage after 63 minutes, Sterling again on the mark when he coolly converted Sane's angled cross from the left.
Dann headed well over from a Palace corner, and City took off Aguero and replaced him with Gabriel Jesus with one eye on next week's Champions League game against Spurs.
McArthur made way for Bakary Sako for Palace, John Stones replacing Sane for City, who removed any lingering doubts about the result when Jesus kept his cool after De Bruyne had played him into the area.