After an entertaining first half ended goalless Juan Mata fired Luke Shaw’s teasing cross into the roof of the net in the 48th minute before Swansea took control.
Andre Ayew restored parity with a headed effort 13 minutes later before Bafetimbi Gomis maintained his superb form with a low shot five minutes later as the Welsh side registered their third successive 2-1 league victory over United.
“It was a very difficult game,” Swansea captain Ashley Williams told Sky Sports. “We had to dig deep.
“Any time you can beat one of the top teams shows our improvement. We keep working every day and every week. We want to be a consistently good side.”
Earlier, Dusan Tadic’s brace and a goal from Graziano Pelle gave Southampton’s stuttering league campaign lift-off with a comfortable 3-0 victory over promoted Norwich City at St Mary’s .
Norwich defender Steven Whittaker was sent off having received two yellow cards within four first-half minutes for professional fouls as the south-coast club registered their first victory of the season.
United’s solid yet unspectacular start to the season had been built on a steely defence which had yet to concede and had restricted opponents to just five shots on target in three matches this season.
Swansea, however, achieved a Premier League double over United last season and had started the season unbeaten, including an impressive opening-day draw at champions Chelsea.
United started brightly, though, with Memphis Depay and Mata sending early warnings of United’s attacking threat.
Swansea began to grow into the game having weathered United’s early attacking raids with Gylfi Sigurdsson firing agonisingly wide before Bafetimbi Gomis curled an effort against the post.
Wayne Rooney netted a midweek Champions League hat-trick but continued his underwhelming domestic form, squandering a host of inviting chances and having fewer first-half touches than every other United player.
After the break Mata fired in before the hosts turned the game on its head.
Ayew headed home Sigurdsson’s cross before the Ghanaian’s stunning pass created space for Gomis to slide home his ninth goal in his last 10 Premier League matches.
The surprise packages of last season having finished seventh, Southampton had been out-of-sorts, taking just two points from three games and failing to score since the opening day.
Sunday’s first half was a microcosm of their season, plenty of possession and chances but a lack of killer instinct in front of goal.
James Ward-Prowse, Fonte, Pelle and the impressive Sadio Mane all went close in a dominant yet fruitless opening 20 minutes before Whittaker was dismissed.
After Pelle stabbed home in the 45th minute Tadic fired home Mane’s cross for a superb second then applied the finishing touch after Pelle’s header was parried into his path two minutes later.
Jay Rodriguez, still making his way back from a long-term knee injury, was brought on after the break and was lively throughout, forcing John Ruddy into a number of saves.
“We haven’t had the start we wanted in the league,” Saints captain Jose Fonte told Sky Sports. “But the most important thing today was getting back to winning games.
“We have massive responsibility now we are seen as a good team so we need to win and that’s the mentality we have to implant in every game.”
Swansea, who have eight points from four games, leapfrogged United, with seven, into fourth. Southampton moved up to 10th with five points while Norwich sit one point adrift in 14th.
Manchester City top the standings going into the international break with a maximum 12 points from four games, three ahead of Crystal Palace.
(Editing by Ed Osmond)