Patrice Evra had put Sir Alex's side in front after 16 minutes when he headed home Robin van Persie's corner at the front post. But Michu equalised for the home side and then the Swans continued to threaten the United goal up until the half-time interval.
The Reds came out with renewed vigour after the restart and created numerous openings to snatch a win, including a van Persie volley that hit the bar and a Michael Carrick header which also struck the woodwork.
Victory wasn't to be, however, as Swansea held out. Nevertheless, United remain four points clear at the top of the league and are guaranteed top spot on Christmas Day.
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