Patrice Evra had given United the lead in the 16th minute as he headed home from a Robin van Persie corner.
But Swansea continued their good record against the top-flight big guns as Michu finished from close range to move clear of Van Persie with his 13th league goal of the season.
David De Gea easily saved weak efforts from Wayne Routledge and Michu, but United halted the hosts' building momentum as they took the lead. Wayne Rooney had already warmed Michel Vorm's hands with a stinging drive before Evra put the visitors ahead, arriving to glance a header in off the far post from a Van Persie corner.
But Swansea recovered well from the blow and Jonathan de Guzman had the ball in the net two minutes later, but the effort was ruled out for Michu blocking off Jonny Evans as the midfielder advanced.
But Michu levelled on the half hour mark. Evra was caught out by De Guzman's clever run and when De Gea could only parry the Dutchman's angled shot, Michu was on hand to tap in.
De Guzman and Routledge were becoming increasingly influential for the hosts and the former should have done better after working his way into the United box on the byline, only to shoot straight at De Gea with several team-mates well placed.
United began to exert some authority and Van Persie came close to putting them back in front as he cleverly diverted a Rooney cross goalwards, only to see his effort cannon away off the bar.
Tempers flared as United frustrations grew. Van Persie was struck on the head by an Ashley Williams clearance after being brought down on the edge of the box leading to some pushing and shoving and a booking apiece for the central protagonists.
Vorm tipped a Michael Carrick header against the bar while Van Persie and Ashley Young were denied by blocks from Williams and Ben Davies, although there was a suspicion of handball about the latter, as Swansea held out.
Source: PA
Source: PA