Two goals late in the first half put United in complete control, and when van Persie added a third – his second of the game – in the 72nd minute a comfortable victory looked on the cards.
Wilfried Bony got one back for the hosts before Welbeck added his second in injury time.
Dutchman Van Persie opened the scoring with a sweetly struck volley before completing his brace with a solo effort from the edge of the penalty area.
In between his two strikes, Welbeck scored from close range after good work from Antonio Valencia.
Bony - Swansea's record signing - got a consolation in the closing minutes with a cool finish but they any hope of a comeback was ended by Welbeck's perfectly executed chip 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, swivelled and emphatically volleyed the ball past Vorm.
And a few moments later, Welbeck doubled their lead, the striker tapping home from a Valencia cross after the Swansea defence 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 their 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 their lead with 18 minutes to play as Van Persie grabbed his second of the match.
The Swansea defence 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.
