Pep Guardiola suffered league defeats home and away to the same opponents in a single league season for the first time in his managerial career as City are left to fight for a top-four place.
Eden Hazard gave Chelsea a 10th-minute lead when his shot deflected off returning City captain Vincent Kompany past keeper Willy Caballero, who should have done better.
Chelsea keeper Thibaut Courtois was also badly at fault when his poor clearance to David Silva set up Sergio Aguero's equaliser after 26 minutes - but Chelsea were back in front before half-time.
They were awarded a penalty after Fernandinho tripped Pedro and even though Cabellero saved Hazard's spot-kick, the rebound fell kindly for the Belgian to score.
City, who stay in fourth, had the better chances in a tense second period, with Kompany's header bouncing back off the bar and John Stones shooting over from six yards in injury time. Chelsea, however, held on for a win that was even more vital given Tottenham's dramatic late comeback at Swansea City.
Chelsea's progress towards the Premier League title has been a tale of almost unbroken serenity since manager Antonio Conte reworked his tactical approach after successive losses at home to Liverpool and away to Arsenal in September.
This was arguably their biggest game since as it followed on from the shock home loss to Crystal Palace and it was against a Manchester City side with the talent and capability to make this night at Stamford Bridge a real test of nerve.
And so it proved as City, with Silva the orchestrator supreme, putting Chelsea's defence and their supporters on edge right until the final whistle.
Chelsea emerged triumphant thanks to the mixture of talent and resilience that has served them so well this season and the celebrations at the final whistle reflected just what a significant night this might prove to be.
Hazard provided the flourishes but manager Conte proved his pragmatism with the introduction of Nemanja Matic for Kurt Zouma at the start of the second half to attempt to lock down the win.
It worked to an extent but Chelsea also enjoyed good fortune as Kompany's header bounced back off the bar and Stones somehow scooped an injury-time chance over the top.
In the final reckoning, Chelsea showed the bloody-minded defiance of champions - and this is the sort of result that could earn them that crown.