Ivanovic scored with a crisp left-footed drive in the 32nd minute at the Etihad Stadium to move Jose Mourinho's side level with City in second place in the Premier League, just two points adrift of leaders Arsenal.
Prior to Monday's game, City had won 19 of their last 20 matches in all competitions and had scored in 61 successive top-flight fixtures at home, going back to a 0-0 stalemate with Birmingham City in November 2010.
However, it was Chelsea who looked most likely to score again, as they struck the woodwork three times going in search of a second, while City struggled to find a way through their well-organised defence.
The victory means Mourinho's team have now won seven of their last eight Premier League outings, and they now sit behind City on goal difference only.
The hosts, with Martin Demichelis in midfield instead of the injured Fernandinho, started much the brighter and created a handful of opportunities in the opening 20 minutes, with Yaya Toure at the heart of everything.
With 10 minutes played, City released Aleksandar Kolarov on the left hand side before the Serbian delivered a dangerous ball across the six-yard box, but a lunging Toure could not quite reach it at the back post.
The Ivorian turned creator eight minutes later after charging into the Chelsea penalty area, he held off the challenge of Nemanja Matic before rolling the ball into David Silva but the Spaniard's flicked effort went just wide of Petr Cech's goal.
David Luiz, starting in midfield for Chelsea, then wasted a free-kick in a good position, and his Brazilian compatriot Ramires did the same shortly after, but the visitors were soon ahead.
Just after the half-hour mark, another swift move from Chelsea saw Ramires’ effort blocked, only for the loose ball to drop to Ivanovic.
The full-back duly lashed a left-footed shot through a crowded penalty area and beyond Hart for his second Premier League goal of the season.
City should have been further behind in the closing minutes of the first half, but Samuel Eto'o fired his shot against the post after excellent work from Eden Hazard.
More poor defending at the start of the second half nearly saw Chelsea double their lead through Willian, and the hosts were again let off the hook when Matic's superb long-range effort crashed against the post.
Hart's goal was leading a charmed life as the visitors hit the woodwork for a third time in the 67th minute. Willian's right-wing corner was met by the head of Gary Cahill, but his effort struck the post and was cleared by a relieved City defence.
Kolarov had City's best attempt of the second half from a 20-yard free kick, but the Serbian saw Cech claw the ball out of the top corner, while Silva spurned a chance to level in the 75th minute.
As the hosts pushed for an equaliser, they could have been reduced to 10 men as Matija Nastasic hauled down susbtitute Oscar on the halfway line, but the defender was only booked - much to the frustration of Mourinho on the touchline.
Stevan Jovetic, who came on for Alvaro Negredo in the second half, then forced Cech into another fine save in injury time as Chelsea held firm in the face of strong late pressure from the hosts.
