Pep Guardiola's men were well below par in the first half but were transformed after the break, with substitutes Jack Grealish and Riyad Mahrez combining for the only goal, scored by the Algerian in the 63rd minute.
Graham Potter's Chelsea came into the game at Stamford Bridge languishing in 10th place in the table but they had the better of the opening 45 minutes.
However, it was a different story in the second half as the visitors found their attacking rhythm, with Guardiola admitting they were "miles better in every department" after the break.
"We struggled a lot in the first half - it was sloppy and slow and without rhythm," the City boss told Sky Sports. "In the second half it was much, much better from minute one.
"The three points are important, definitely. The players know it. For a long time we were at the top. But we try, we have the FA Cup and Carabao (League) Cup and then Old Trafford. But it's important for these games, a win today."
Chelsea, already without a clutch of senior players including Mason Mount and Reece James, lost forward Raheem Sterling to an apparent hamstring injury in the opening minutes, replaced by Pierre-Emerick Aubameyang.
The home side were first to threaten seriously but City defender John Stones came to the rescue in the 16th minute as Christian Pulisic bore down on goal, producing a perfectly timed sliding tackle to deny the US forward.
The American was injured in the challenge and was forced off minutes later, compounding Chelsea's problems.
Potter is under increasing pressure - Chelsea now have just one win in their past eight league games and remain 10 points off the top four.
The Blues manager, whose side face City again in the FA Cup third round at the Etihad on Sunday, said he was proud of his team's performance, especially after losing Sterling and Pulisic so early.
"It's tough at the moment I must admit and I feel for the boys," he said.
"We have to stick together. It was disappointing to lose those guys but the players on the pitch and the ones that came in gave everything and that's all you can ask for."
