Two eye-catching goals from Robin van Persie, either side of an effort from Danny Welbeck, gave the Red Devils a commanding three-goal lead at the Liberty Stadium.
Micheal Laudrup's men pulled a goal back through league debutant Wilfried Bony but Welbeck rounded off the win for the visitors with a well-taken second goal scooped over Michel Vorm.
Following his performance for England against Scotland on Wednesday, Moyes named Wayne Rooney on the bench, as speculation over his future at Old Trafford continued.
The United boss instead preferred to use Van Persie as a lone striker, with Ryan Giggs and Welbeck providing support.
Swansea boss Michael Laudrup gave a debut to Jose Canas, although Bony had to settle for a place on the bench.
The better of the early chances fell to the visitors - Phil Jones seeing an early volley tipped over the crossbar by Vorm while Giggs wasted two good opportunities with a rare lack of composure.
But United’s early dominance was turned into goals just after the half hour with two in as many minutes.
First, Van Persie broke the deadlock with a piece of excellent opportunism. The Dutchman controlled a ball from Giggs on his chest, swiveled and emphatically volleyed the ball past Vorm.
And a few moments later, Welbeck doubled the lead, as the striker tapped home from a Valencia cross after the Swansea defense had failed to clear Patrice Evra’s dangerous cross.
With his side chasing the game, Laudrup introduced Ivory Coast striker Bony and Pablo Hernandez at the start of the second half, and slowly Swansea worked its way back into the game.
Their clearest sight of goal came just before the hour when a loose ball fell to Canas on the edge of the penalty area, but his long-range effort sailed harmlessly over the crossbar.
United, having introduced Rooney, extended its lead with 18 minutes to play as Van Persie grabbed his second of the match.
The Swansea defense backed off as he bore down on goal, before the Dutchman jinked away from a Chico Flores tackle and fired the ball beyond Vorm's outstretched dive into the top corner.
Bony capped his Premier League debut with a goal, the Ivorian stroking the ball home first time from inside the penalty area after excellent work from Hernandez, but his strike with eight minutes to play ultimately mattered little.
Welbeck added his second in the final minute of injury time, the striker calmly lifting the ball over Vorm from a tight angle despite close attention from Chico, to seal the win for United.