In a significant day in this season's title race, in-form England striker Wayne Rooney grabbed both goals for United at Old Trafford to take his tally to nine goals in six matches.
City was sunk by an 83rd-minute goal by Swansea striker Luke Moore, losing the lead to its fierce rival by the end of a round of fixtures for the first time since Oct. 15.
United has won seven of its past eight matches in a familiar late-season charge that left manager Alex Ferguson confident of the northwest giants extending their record haul of English titles to 20.
"We've turned it around and credit to the players for that. They have shown great resilience and determination," Ferguson said. "We won't get nervous, we have the experience, we will keep on playing and it is good to see that type of temperament."
City has failed in recent months to recapture the free-flowing style of football that allowed it to blitz allcomers in the opening half of the season.
Although its home record remains perfect, City has lost four times away from Etihad Stadium and it could been worse had Swansea forward Scott Sinclair's penalty not been saved by goalkeeper Joe Hart.
"The lead can change every game but it is important to start winning and scoring again," City manager Roberto Mancini said. "All the players at this moment are tired, but we have a lot of energy to go top again and there are 10 games to play."
Both teams lost the first legs of their Europa League last-16 matches on Thursday but it was United which reacted better, despite West Brom coming into the game on the back of three wins in a row — the most recent a 1-0 victory over Chelsea that ultimately cost Andre Villas-Boas his job.
Rooney moved to 20 league goals this season — and 26 in all competitions — after touching in a shot by Javier Hernandez in the 35th minute. He smashed home a 71st-minute penalty to seal the points after Ashley Young had been tugged back.
"We are plugging away," Rooney said. "We have to keep going and keep picking up the points to get over the line. We know what we have to do and hopefully it will be enough."
Of all United's strikers, Rooney looks most likely to fire the team to the title.
"If Wayne can get to 40 goals, I'll be absolutely delighted because we will be in business then," Ferguson said.
City was caught napping for Swansea's goal when Yaya Toure was caught in possession by Gylfi Sigurdsson. Wayne Routledge crossed from the right and Moore headed past Hart.
Micah Richards thought he had earned City a point, which would have left it top of the table on goal difference, when the right back headed in but he was ruled offside.
"We were clearly the better side throughout the game," said Swansea manager Brendan Rodgers, whose team looks to be safe at 11th in the standings — 14 points above the relegation zone in its first season in the Premier League.
"Today is all about the Swansea City players and I can't think of better days."
City has a tougher run-in than United, on paper at least, with games to come against Chelsea at home and Arsenal away.
Neutrals will hope the title race is still on when the Manchester rivals meet in the derby on April 30, both sides' third-last game of the campaign.
In the day's other match, bottom team Wigan earned a precious point in the fight against relegation with a 1-1 draw at midtable Norwich.
Victor Moses' 68th-minute equalizer cancelled out Norwich's opener from Wes Hoolahan in the 10th.
Wigan is a point behind second-bottom Wolverhampton and two adrift of safety.