City had won all 11 of their previous home league games this season, losing only to European champions Bayern Munich on home turf, while it was the first time they had failed to score at the Etihad in the league since November 2010.
The hosts had been seeking a ninth successive league win, but instead Jose Mourinho’s were left to celebrate a home-and-away double, having beaten City 2-1 at Stamford Bridge in October’s reverse fixture.
Aleksandar Kolarov whipped a cross across goal that narrowly eluded , who then clipped a shot just over from the edge of the box before striding to the byline and teeing up David Silva to sweep an effort narrowly wide.
Edin Dzeko curled off-target after giving David Luiz the slip, but City’s gung-ho advances left them exposed at the back and Joe Hart had to save from Ramires in the 27th minute after a cut-throat counter.
It was a reminder of the visitors’ pace on the break and five minutes later another raid saw them go ahead.
Vincent Kompany did well to block from Ramires, who had been picked out by Eden Hazard, but as the ball rolled to the edge of the box Ivanovic met it with a devastating left-foot drive that flew into the bottom-left corner.
With City rattled, the irrepressible Hazard worked Hart with a low strike before skipping past Pablo Zabaleta and crossing for Samuel Eto’o, who could only slam against the crossbar from a narrow angle.
An exchange of chances saw both sides go close early in the second half, with Toure drilling a shot a whisker wide at one end before Nemanja Matic clipped the post with a 30-yard thunderbolt at the other.
The hosts responded, Dzeko miscueing from a Zabaleta centre, while Silva drew a fine one-handed save from Cech with a dipping free-kick before clattering the ball wide from a Kolarov cross.