The Premier League champions came into Saturday's match with just one win from their last six matches in all competitions.
Wilfried Bony made the most of frailties among Manuel Pellegrini's team by handing Swansea an early lead at the Etihad Stadium, but recalled forward Stevan Jovetic restored parity after 19 minutes.
Below-par performances from midfield talisman Toure have often defined City's malaise this term, but the Ivory Coast international showcased his trademark combination of power and poise to net decisively 17 minutes into the second half.
Swansea boss Gary Monk used his substitutions shrewdly to craft a dramatic finish, with replacements Bafetimbi Gomis and Jonjo Shelvey agonisingly close to securing a late share of the spoils.
But 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 contributed to Jovetic starting 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 eight minutes.
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.
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 lent a feisty edge to an opening period that threatened to boil over when Bartley and Aguero clashed once more.
Lukasz Fabianski brilliantly tipped a venomous Gael Clichy long-range strike onto the post after 39 minutes, and Pellegrini's men resumed on the front foot after the interval, but a slick exchange of passes between Bony and Jefferson Montero forced Hart into a fine save from the latter.
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 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 wide with Hart beaten.
City will certainly require far more poise in Tuesday night's must-win UEFA Champions League clash with Bayern Munich.
