Roberto Mancini's side drop vital points in the title race following a lacklustre performance at the Liberty Stadium as former Aston Villa striker heads in late winner
Luke Moore's strike seven minutes from time secured victory for Brendan Rodgers' side, as Roberto Mancini's men now trail United by a single point.
The Welsh outfit had the opportunity to grab the lead in the first half but Joe Hart made a brilliant save from a Scott Sinclair penalty.
City lacked their usual control and dynamic play as David Silva and Mario Balotelli failed to spark the visitors into life as they barely threatened the goal of Michel Vorm.
The Swans play throughout the game was rewarded when Moore rose the highest to head Wayne Routledge’s cross into the net to give the home side the win as Micah Richards had a late effort disallowed for offside.
Rodgers’ side settled into the game from the off and started to pass the ball around with their usual swagger despite the quality of their opponents. This early pressure nearly paid off as a good move presented Routledge with a good chance but he put his shot over the bar.
Swansea were given a great early chance to take the lead when Routledge was brought down in the area by Hart and referee Lee Mason pointed to the spot. Sinclair took a stuttered run up before firing a tame penalty which was easily saved by Hart palming the ball away from goal.
City were unable to create any meaningful chances in the first-half as the Swans controlled the midfield while Ashley Williams and Steven Caulker marked Mario Balotelli out of the game. Their best efforts came for set-pieces and long-range shots neither of which troubled Michel Vorm.
Roberto Mancini’s side responded to Swansea’s dominance in the first-half by coming out strongly after the interval and threatened immediately through Sergio Aguero who beat his marker before firing a vicious shot at goal which went just wide of the post.
The Swans, however, came back at the Premier League leaders and came close to grabbing the lead when Danny Graham fired a ball across the face of goal which needed only a touch to be turned into the net but unfortunately for the home-side no-one was on hand to tap the ball home.
City were incensed when Balotelli was felled seemingly inside the area by Joe Allen but referee Mason waved play on, leading to vigorous protests from De Jong and Yaya Toure on the field and Mancini on the touchline.
The visitors were beginning to get a grip on the game with Silva becoming more involved and it was his ball who found Richards in the area who twisted and turned before shooting a powerful effort at goal which was well saved by Vorm.
Swansea however remained resilient in defence and were rewarded for their play when they took the lead in the 83rd minute when Moore headed Routledge’s cross into the top corner past a stranded Hart following a mistake by Gael Clichy.
Richards thought he had won City a point late-on when he headed the ball into the net as Vorm came and missed Clichy’s cross but as he celebrated, official Sian Massey raised her flag for offside much to the delight of the home crowd.
Mancini’s side pressed for an equaliser but were unable to find a way through the Swansea defence as they dropped vital points in the Premier League title race as the home-side held on for a deserved win at the Liberty Stadium.