The champions took the lead as Javier Hernandez reacted quickest in the area and fired home.
Swansea came out strongly in the second half and equalised just four minutes after the interval.
Nathan Dyer fired in a cross and Michu did well to get to the ball before Phil Jones and flick the ball past David de Gea in the United goal.
United were not to be denied victory on Sir Alex Ferguson’s special day and found a winner on 87 minutes. The Swansea defence failed to clear Robin van Persie’s corner and centre-back Rio Ferdinand lashed home his first goal in five years.
Elsewhere, the fight for survival will now be between Sunderland, Aston Villa and Wigan. Norwich secured their safety this season with a 4-0 victory over West Brom at Carrow Road. Goals from Robert Snodgrass, Grant Holt, an own goal from Gareth McAuley and a Jonny Howson strike were enough to secure the Canaries future in the top flight.
Newcastle also ensured their survival with a 2-1 victory over rock-bottom QPR. The home side took the lead on 11 minutes through a Loic Remy penalty. Seven minutes later the Magpies equalised through a Hatem Ben Arfa penalty kick.
The visitors took the lead through striker Yoan Gouffran. It was not all plain sailing for the Magpies though, as they were reduced to ten-men when keeper Rob Elliot received a second yellow card. Newcastle hung on to three vital points though.
Sunderland and Southampton drew 1-1 at the Stadium of Light. The Black Cats took the lead on 68 minutes, as Phil Bardsley fired home. The Saints equalised on 76 minutes through midfielder Jason Puncheon.
The draw saw the home side slip to fourth from bottom, while the visitors are now safe due to their superior goal difference.
Other Premier League results:
Stoke 1-2 Tottenham
Everton 2-0 West Ham
Fulham 1-3 Liverpool
Who will join Reading and QPR in the Championship next season?
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