City had won their previous 11 league fixtures at the Etihad this season - and had scored at least in their last 61, stretching back to November 2010 - but their fortress was raided by their title rivals on Monday night.
Jose Mourinho masterminded a narrow success that drew his Blues side level with City on 53 points - two behind leaders Arsenal - with 14 games left to play.
Chelsea were unfortunate not to win by a larger margin. After Ivanovic lashed the visitors in front from the edge of the area on 32 minutes, they hit the woodwork three times through Samuel Eto'o, Nemanja Matic and Gary Cahill.
David Silva threatened to snatch an equaliser as City pushed hard in the final stages, but Manuel Pellegrini could have few complaints about the result on a night when Chelsea displayed discipline and no shortage of skill, with Eden Hazard their stand-out performer in attack and Matic impressing in a central role.
City were without midfielder Fernandinho. The Brazilian was forced out injured, meaning Martin Demichelis moved into midfield and Matija Nastasic returned in defence. Alvaro Negredo was restored up front for the injured Sergio Aguero.
Chelsea left playmaker Oscar on the bench as they adopted a more defensive approach, with David Luiz and Matic replacing him and John Obi Mikel.
Silva turned the ball the wrong side of the post early on after good work by Yaya Toure, but City looked increasingly open at the back - Willian somehow squandering a four-on-two situation - and were punished just after the half-hour mark.
Ramires saw a shot blocked by Vincent Kompany and the ball fell to Ivanovic, who fizzed in a left-footed shot into the bottom corner giving Joe Hart no chance.
Referee Mike Dean booked two players from either side before the break as the match became more competitive, and Chelsea almost doubled their advantage on 44 minutes, only for Eto'o to see his shot from a tight angle come back off the crossbar after Hazard's pass had slid across the area.
Toure drilled wide from Dzeko's lay-off early in the second half and after Matic had seen a long-range effort clatter the bar, Stevan Jovetic replaced Negredo in the City attack on 57 minutes.
Ten minutes later, Chelsea had again tested the frame of the goal via Cahill's header from a Willian corner.
Silva had two sights of goal but Cech was equal to the first - a free-kick that was expertly pushed away - and the second moments later was poked wide under pressure.
On 87 minutes, Mourinho was enraged when last defender Nastasic was only booked for hauling down substitute Oscar on the halfway line, but it was not a pivotal decision.
Cech was called into action once more in stoppage time to deny Jovetic's fiercely-struck effort, while Nastasic miskicked when left unmarked in the area at the death.
Chelsea hung on for a victory which saw them become the first team to do the double over City since Everton in 2010/11 - and they could now overtake both Pellegrini's team and leaders Arsenal if results go their way on Saturday.