Javier Hernandez struck in the 11th minute, sliding home his fifth goal of the season from Ryan Giggs' pass after Angel Rangel had carelessly given the ball away on the edge of his own box, to give United their first win at Swansea at the ninth attempt.
Scott Sinclair missed Swansea's best chance when he miscued with an empty goal at his mercy, but, despite some bright attacking moments, the Welsh side struggled to create against a well-organised United rearguard as they suffered a first home defeat of the campaign.
Swansea's brightest moments had come during the early stages of the contest, with wingers Nathan Dyer and Sinclair proving their primary attacking threat with several raids down their respective flanks.
But all the promise of an energetic home start was undone when Rangel's lazy pass gave the ball away to Giggs and the Welshman had little trouble finding Hernandez, who beat Michel Vorm from close range.
The goal briefly rocked the hosts back on their heels but a wonderful jinking run from Dyer, which saw the winger beat Wayne Rooney, Giggs and Patrice Evra along the way, soon had the home crowd on their feet.
And Swansea should have been level after 22 minutes as a deflected shot from Wayne Routledge found Sinclair with the goal at his mercy, but the ex-Chelsea man badly missed his kick and the ball deflected to safety off his standing leg.
Despite those bright moments Swansea were finding it difficult to get into their usual passing rhythm as United pressed high up the field and penned them in.
The champions' increasing comfort saw Rooney, who excelled playing just off Hernandez, flash a long-range drive wide, while Giggs curled a free-kick narrowly off target.
Hernandez should have doubled the lead to reflect the visitors' superiority right on half-time, but he opted to square the ball after Rooney's wonderful control and weighted pass had put him in on goal.
With United in such a position of control Swansea boss Brendan Rodgers summoned precocious midfield talent Joe Allen, who was nursing a calf problem, from the bench in place of Routledge in an effort to change the pattern of the game.
And Allen had an instant impact as he teed up Sinclair for a shot that David De Gea tipped round his post.
But United remained dangerous and Hernandez again failed to capitalise on more magic from Rooney as he miscontrolled his strike partner's superb ball towards the six-yard box.
But Swansea had started the second half with more verve and vigour and it took a brave defensive block from Nemanja Vidic to prevent Ashley Williams getting on the end of Garry Monk's goalbound header.
Michael Carrick and Rio Ferdinand blocked on-target efforts from Danny Graham and Williams as the hosts continued to push forward in search of a leveller.
But having weathered that brief storm United took a stranglehold on possession, with Carrick's assuredness on the ball taking the sting out of Swansea's attacking ambitions.
Rooney had a chance to cap a fine individual display late on but chipped over the bar after getting a lucky ricochet from a Williams block, before Phil Jones clipped the post and Nani fired wide as time ran out.