The Ivorian linked up well with Fernandinho to help Manuel Pellegrini's men come from a goal down to beat Garry Monk's men at the Etihad Stadium
The Premier League champions came into Saturday's match with just one win from their last six matches in all competitions, and face Bayern Munich on Tuesday with their Champions League hopes hanging by a thread.
Wilfried Bony made the most of frailties among Manuel Pellegrini's team by handing Swansea an early lead at the Etihad, but recalled forward Stevan Jovetic restored parity after 19 minutes.
|VIEW FROM THE ETIHAD|
|By Paul Macdonald
Yaya Toure's second half goal gives City a vital victory, but the deficiencies that plague this team just won't go away.
Swansea gave as good as they got throughout, and in the closing stages both Bafetimbi Gomis and Jonjo Shelvey came agonisingly close to equalising, the latter a deflected free kick that looked for all the world to be finding the top corner, only to swerve wide at the last moment.
Bayern Munich will analyse that defensive display and know how to hurt City - they are not an organised unit and it is all too easy to pass through them. Despite this win, that will keep Manuel Pellegrini awake tonight.
Swansea boss Garry Monk used his substitutions shrewdly to craft a dramatic finish, with replacements Bafetimbi Gomis and Jonjo Shelvey agonisingly close to securing a share of the spoils.
Pellegrini's men remain eight points behind leaders Chelsea, but close to within a point of Southampton before Ronald Koeman's second-placed team travel to Aston Villa on Monday.
Captain Vincent Kompany returned from a calf problem for the home team but a similar complaint for Edin Dzeko meant Jovetic started in attack.
Shelvey had to settle for a place on the Swansea bench following a one-match suspension, while Nathan Dyer returned to the starting XI and the winger played a key role in the opening goal after nine minutes.
As it has been so often this season, the champions' defensive vulnerability was on full display when Bony charged onto a return chipped pass from Dyer to coolly slot past Joe Hart.
There was little conviction in City's response until Toure released Jesus Navas and the Spain winger fizzed in an excellent low right-wing cross to leave Jovetic with a simple finish to equalise.
Swansea defender Kyle Bartley was booked for a robust 23rd-minute tackle on Sergio Aguero before an aerial challenge between Kompany and Neil Taylor left the visiting full-back with a cut eye - two incidents that gave a feisty edge to an opening period that threatened to boil over when Bartley and Aguero clashed once more.
There was a pleasing ebb and flow to the second half and Pablo Zabaleta streamed forward on the overlap to collect Navas' pass but blazed over left-footed having cut inside Ashley Williams.
Fabianksi was unsighted when Toure drilled goalwards in the 61st minute - he got down well to save but had no chance when the same player put Pellegrini's side ahead little more than a minute later.
Tom Carroll was careless in possession, allowing Samir Nasri to find Fernandinho in the Swansea box and his midfield partner Toure charged onto a perfectly weighted backheel to prod home.
City then went for the kill and Jovetic curled over after Aguero’s shot was blocked.
Navas became the latest attacker to be frustrated by Fabianski but late nerves afflicted City as Gomis spurned an 87th-minute one-on-one and Shelvey's deflected free-kick spun agonisingly wide with Hart beaten.
City will certainly require far more poise in Tuesday night's must-win Champions League clash with Bayern Munich.