After a competitive first half, the Red Devils dominated proceedings in the second period and hit the bar twice, despite Wayne Rooney producing a very poor display
The league leaders opened the scoring through Patrice Evra in the first half, the left-back heading in a corner, but the hosts equalised after good work from Wayne Routledge and Jonathan de Guzman, Michu tapping in, and they held on for a point, despite the visitors dominating the second half and hitting the bar twice.
Michael Laudrup made just the two changes from the side that lost to Tottenham in their last fixture, as Michel Vorm and Kemy Agustien came in for Gerhard Tremmel and Ki Sung-Yueng.
Sir Alex Ferguson also made two changes, both coming at the heart of his defence, with Rio Ferdinand and Chris Smalling making way for Nemanja Vidic and Jonny Evans.
The match had been built up as a face-off between the division's two top scorers, Robin van Persie and Michu, and the Spaniard was the first to get through on goal as he broke away from Evans inside 10 minutes but his wild effort went well wide.
United, however, soon had the lead as Van Persie whipped in an inviting corner from the right and Evra rose to bag his third goal of the season, diverting the delivery into the far side of the net.
Both sides continued to create chances but it was Swansea who grabbed the next goal as De Guzman fired at David de Gea after a clever Routledge pass into the area, only for Michu to tap in for his 13th league goal of the season.
The equaliser proved a massive confidence boost as the hosts pressed for a second and, after some awful defending by Tom Cleverley in the right-back position, De Guzman stole the ball and shot, though De Gea caught his effort at the near post.
United created a few attacks late on in the half but, with Antonio Valencia and Ashley Young both having poor matches, their supply from the flanks was poor and the two teams went into the break level.
The visitors began the second half in improved fashion but Routledge was soon causing trouble again as he produced one stepover before shooting straight at De Gea, while Young had an effort blocked at the other end within seconds.
United took control of the midfield around the hour mark, but with the likes of Wayne Rooney and Young looking well off the pace, supply to Van Persie and substitute Chicharito was dreadful.
However, Rooney soon produced one excellent to delivery, but the crossbar came to Vorm's rescue after Van Persie's flick, while the Dutchman soon set up Michael Carrick for a chance of his own, but the midfielder's header came back off the woodwork too, seconds after the disappointing Rooney was hooked for Ryan Giggs.
Sir Alex's men had their hosts under significant pressure for the final minutes but, with the final ball lacking, Swansea held on for a very commendable point, their first ever one in the Premier League against United.