The Swans were denied a point late on at champions Manchester United on a day where Michael Laudrup's side were guaranteed a top-half finish in the Barclays Premier League.
United gave manager Sir Alex Ferguson a winning send-off as he took in his last home game in charge before his retirement.
Javier Hernandez struck the crossbar before giving United the lead, but it was the Swans who had the better of the chances in the second half.
And when Michu struck his 22nd goal of the season the Swans took great heart and went close to taking the lead on a few occasions.
But Rio Ferdinand hammered home from close range following a corner in the 87th minute to give the hosts all three points.
Results elsewhere, however, ensured the Swans would finish in the top-half of the table regardless of results on the final day of the season.
With the Swans back in the changing rooms, the celebrations took over with Sir Alex Ferguson lifting the Barclays Premier League trophy aloft.
The Swans made five changes to the starting XI.
Gerhard Tremmel started in goal as Michel Vorm missed out due to the injury he picked up against Wigan in midweek but the big news was Neil Taylor being handed his first league start since September.
Chico Flores was back in defence as Garry Monk moved to the bench and Nathan Dyer was also recalled while Michu led the line as he returned from injury.
Manchester United, determined to give Sir Alex Ferguson a fitting send-off in his final home game in charge, also said farewell to Paul Scholes - the midfielder being handed a start in his last at Old Trafford before his retirement.
There was no place in the matchday squad for Wayne Rooney, with Ferguson opting for Javier Hernandez and top scorer Robin van Persie up front.
Manchester United: David de Gea, Phil Jones, Nemanja Vidic (capt), Rio Ferdinand, Patrice Evra, Danny Welbeck (Antonio Valencia, 67), Michael Carrick, Paul Scholes (Anderson, 67), Shinji Kagawa, Javier Hernandez (Ryan Giggs 76), Robin van Persie.
Subs: Anders Lindergaard, Jonny Evans, Alexander Buttner, Tom Cleverley.
Swans: Gerhard Tremmel, Dwight Tiendalli, Chico Flores, Ashley Williams (capt), Neil Taylor (Ben Davies, 67), Leon Britton, Jonathan de Guzman, Wayne Routledge, Nathan Dyer, Pablo Hernandez (Kemy Agustien, 87), Michu (Angel Rangel, 76).
Subs: David Cornell, Garry Monk, Itay Shechter, Roland Lamah.
Old Trafford was fully charged up prior to kick off, with both sides holding a guard of honour for Sir Alex Ferguson.
Then it was onto the game, with the atmosphere not dampened despite the rain dropping down onto the pitch.
There was just five minutes on the clock when the home crowd were on their feet as Javier Hernandez's shot rattled the crossbar and bounced out of danger following a swift Manchester United attack.
The Swans recovered, and Pablo Hernandez had a good chance to break the deadlock after Michu found him on the left during a counter attack, but the Spanish winger cut inside before scuffing his shot wide.
Javier Hernandez then threatened at the other end, meeting a cross only to see his header blocked by Chico Flores.
Gerhard Tremmel then had to deal with Patrice Evra's low drive before a neat counter from the Swans created some space for Nathan Dyer, but he fired over.
United kept probing, and a fine flicked pass from Robin van Persie found Javier Hernandez at the edge of the box, but his volley on the turn failed to trouble Tremmel as it flew into the home support behind the goal.
Then, in the 39th minute, United got themselves in front.
After being awarded a free-kick, van Persie's delivery just evaded Pablo Hernandez and deflected off Ashley Williams into the path of Javier Hernandez, and he made no mistake from close range.
Tremmel then did well to palm out a powerful van Persie shot before the Dutch striker scooped another effort wide with his right foot.
Half-time: Man Utd 1 Swans 0
The Swans made a superb start to the second half, levelling in the 49th minute.
After Williams had gone close with a header, Dyer's curling cross was met by the onrushing Michu, who volleyed home with a powerful effort.
The Swans continued their excellent start, Michu linking up with Routledge to send the winger in on goal, but Rio Ferdinand did just enough to ensure Routledge's effort spun off the surface and inches wide of the far corner.
Then, during an excellent counter, Michu ran on and slipped in a pass for Dyer, who then found Pablo Hernandez, and when he beat his marker he teed up a shot. But David de Gea was alert and swatted the effort for a corner.
From the resulting delivery, Chico controlled and saw his shot deflect back to him off Phil Jones before Evra then got in front of his second effort.
With 67 minutes on the clock, Michael Laudrup made his first change with Ben Davies replacing Neil Taylor.
But it was United who next went close, sub Anderson - on for Scholes - volleying just past the post from Tremmel's punched clearance.
Laudrup made another change - Angel Rangel coming on for Michu, which meant Dyer led the line.
Pablo Hernandez then went close to giving the Swans the lead before he was replaced by Kemy Agustien with three minutes.
But just seconds later, United took the lead with Rio Ferdinand hammering home from close range after the Swans failed to clear a corner.