Javier Hernandez's early strike ensured Manchester United kept up the pressure on league leaders Manchester City.
Angel Rangel's error allowed Ryan Giggs to cross for Hernandez, who tapped in after 11 minutes.
United went on to control the first half but Scott Sinclair wasted an excellent chance for the home side, miscuing a shot in the 22nd minute.
Swansea briefly threatened after the break before Phil Jones hit a post late on as United closed the game out.
But this was far from a vintage United performance as they once again struggled to find cohesion in the final third.
Swansea penetrated well down the flanks, with Nathan Dyer giving Patrice Evra a particularly uncomfortable match before the Frenchman was substituted early in the second half after collecting a yellow card.
But the killer edge belonged to United, and more specifically Hernandez as the striker again showed his appetite for away goals by scoring his 13th away from Old Trafford.
United needed an immediate response after rivals Manchester City earlier opened up an eight-point lead at the top of the table, and 11 minutes was all it took to stamp their authority at the Liberty Stadium.
Swansea's passing philosophy has won plaudits in their debut Premier League season, but their willingness to pass in all areas proved to be their downfall.
Giggs intercepted Rangel's weak square pass on the edge of Swansea's penalty area, and the Welshman's precise low cross to Hernandez allowed the Mexican to stab home his fifth goal of the season.
But the lead should have been cancelled out 10 minutes later when Sinclair was guilty of a horrendous miss.
Wayne Routledge, starting in midfield as Joe Allen's slight knock relegated him to the bench, delivered a perfect far-post cross but Sinclair fluffed his shot from six yards and United cleared.
Giggs and Michael Carrick, making his first Premier League start this season, began to control midfield, benefiting from far too much time and space, but Evra's low drive was all that troubled Michel Vorm in the Swansea goal before the interval.
United's unchallenged dominance in midfield had obviously caught Brendan Rodgers's eye as the Swansea manager's response was to replace Routledge with Allen for the second half.
And the young Welshman was immediately involved as he set up Sinclair, whose shot forced David de Gea to save at his near post.
United broke with trademark fluidity from the resulting corner but poor control from Hernandez cost him the chance to double the lead.
Swansea, who had never lost in their eight previous meetings with United at home, showed greater urgency after the break, and the visitors were indebted to Vidic's tenacity in the air to clear twice in his own six-yard box.
Carrick's despairing lunge then blocked Danny Graham's goalbound effort as the home side's pressure grew in the final 20 minutes.
United's only second-half attempts on target were two speculative long-range Rooney efforts until a semblance of control returned to their play in the final 10 minutes.
Another slick counter-attack freed Phil Jones down the right, but his low effort across the face of goal clipped the far post after the slightest of touches from Vorm.
That move was the closest United came to reaching anywhere near top gear, but they did enough to end Swansea's unbeaten home record this season as they closed out for a victory that takes them back to within five points of Premier League pacesetters City.
Swansea manager Brendan Rodgers: "Our squad is very thin but you could see the confidence in the team in the second half.
"In the first half we were nowhere near our level and we made it too comfortable for Manchester United. In the end we're disappointed not to get a point.
"But there's a great confidence in the team and if we carry on this season like we played in the second half then we'll give ourselves a chance."
Manchester United manager Sir Alex Ferguson: "We didn't really look as though we were going to lose a goal.
"In the past when we have won titles we have won four or five games 1-0. Michael Carrick was fantastic and we spoke to Phil Jones this morning about how he felt after the busy week he has had.
"But he wanted to play and that's not a surprise because he's got a lot of energy. He was magnificent in the second half."