Summer signing Gylfi Sigurdsson scored Swansea’s winner with just under 20 minutes remaining, having also teed up Ki Sung-yueng’s first-half opener before Wayne Rooney acrobatically levelled for United early in the second half.
United suffered a blow midway through the first half when debutant Jesse Lingard was forced off due to injury and was replaced by Adnan Januzaj.
Swansea took the lead with the first real chance of the game on 28 minutes when Sigurdsson rolled a pass across the edge of the area and Ki, who spent last season on loan at Sunderland, placed a low shot beyond David de Gea’s reach.
Louis van Gaal brought Nani on for Javier Hernandez at half-time, ditching his 3-5-2 formation in the process, and saw United equalise within eight minutes of the restart.
Juan Mata’s corner was helped on at the near post by Phil Jones and newly-appointed captain Rooney’s overhead kick from close range flew into the net.
Rooney struck the outside of the post with a 25-yard free-kick on 66 minutes before Swansea regained the lead on 72 minutes when Sigurdsson finished after Wayne Routledge’s mis-kicked shot dropped into his path.
United substitute Marouane Fellaini’s only contribution was to escape censure for flicking out an arm at Ki as Swansea comfortably held out to claim a first league win at Old Trafford after beating the Red Devils in the FA Cup third round last season.
A wave of noise greeted Van Gaal and his players as they emerged from the tunnel to signal the start of a new chapter in United's rich history.
Any of the 75,339 fans inside Old Trafford expecting a lightning start to this Premier League season were left disappointed as a dull opening 15 minutes ensued.
Swansea did manage to test United's new-look three-man defence on a couple of occasions.
Routledge raced on to Jonjo Shelvey's through-ball, but De Gea came to the rescue. Routledge then gave Jones the slip down United's right, but his cross was cut out just before Wilfried Bony could convert.
Rooney led by example as he tried to drag United into the game. The new United captain tore around the pitch, but his two first-half attempts on goal were weak.
Lingard, making his full United debut, was replaced by Januzaj after injuring his right foot in a challenge with Ashley Williams.
Finally, with 24 minutes on the clock, Swansea goalkeeper Lukasz Fabianski was tested for the first time when he saved Jones' blistering shot after a clever one-two with Mata.
Then, four minutes later, Old Trafford fell silent as Swansea took the lead. Tyler Blackett was drawn out of position by Nathan Dyer, who laid the ball off to Sigurdsson and the Iceland international squared to Ki, who fired the ball past a diving De Gea.
Bony played an important part in the goal, blocking Jones as he rushed out to challenge the South Korean goal scorer.
Van Gaal sought to inject life into his team at the interval by bringing Nani on for the ineffective Hernandez, who started up front ahead of the unfit Robin van Persie.
Within eight minutes of the restart the hosts were level. Jones climbed well to flick on Mata's corner and Rooney acrobatically swept the ball past Fabianski.
Raucous celebrations ensued on the United bench, but Van Gaal simply recorded the goal on his notepad and looked up with a blank expression.
Neil Taylor went in to the book for hacking down Januzaj and Mike Dean also issued a caution to Shelvey for kicking the ball away following a late tackle on Mata.
Rooney turned Williams with an exquisite bit of skill, but the Swansea captain recovered to deny the England striker.
Rooney then came agonisingly close to putting United ahead with a brilliant free-kick. After being hacked down by Jordi Amat on the edge of the area, the United captain curled a shot up and over the wall, but the ball struck the frame and bounced wide.
In the 72nd minute, Routledge mis-hit Jefferson Montero's cross. Fortunately for Swansea, the ball fell at the feet of Sigurdsson, who bundled the ball past De Gea.
United had a late penalty appeal turned down as Dean waved play on and Swansea held on for their first league win at Old Trafford.