Defender Patrice Evra headed United ahead from a Robin van Persie corner on 16 minutes but the league's top scorer Michu netted his 13th goal of the campaign, moving the Spaniard one ahead of Van Persie, to level just before the half-hour.
United, who hit the woodwork twice in the second half through Van Persie and Michael Carrick, hold a four-point advantage over Manchester City heading into a busy holiday programme.
United have 43 points from 18 games, followed by City on 39 and four clubs on 30. Chelsea have the chance to regain third place with victory at home to Aston Villa in the late kickoff (1600).
"We absolutely battered them and we are very unlucky," United manager Alex Ferguson told Sky Sports.
"It should have been a six point gap (over City) today and that is the disappointment."
Swansea welcomed back goalkeeper Michel Vorm after he missed 10 games with a groin injury and the Dutchman was soon called into action to beat away a Wayne Rooney drive before he quickly followed up to deny Ashley Young.
However, from the resulting corner Evra met Van Persie's delivery with a glancing header that eluded Leon Britton who was stationed on the goalline.
United have scored 10 goals from corners in the Premier League this season, more than any other side.
Their lead lasted 13 minutes with United's defence, welcoming back Serbian Nemanja Vidic for his first start since September, unable to stop Michu pouncing from close range.
Wayne Routledge was the creator for the equaliser, his well-timed pass playing in Jonathan de Guzman and when David de Gea failed to hold his well-struck low shot, Michu was on hand to follow up.
United had the better of the second-half although the hosts, again displaying the attractive attacking verve that has won Michael Laudrup's side many admirers this season, were also a threat until tiring in the closing stages.
Van Persie's mishit volley struck the crossbar before Vorm came to Swansea's rescue when he tipped Carrick's glancing header on to the woodwork.
Young was then then denied by a brave block as United pushed hard for a victory that was to elude them.