For David Moyes, this was just about as good a start as he could have dared to imagine. He will have appreciated the loud adaptation of the old Slade tune Cum on feel the Noize from the end where Manchester United's supporters serenaded their new manager. The photographers got their shot of him punching the air and Moyes must think anything is possible if he can keep Robin van Persie in this form.

The new United looks very much like the old one. Van Persie has carried on seamlessly from where he left off last season, adding two more goals to his collection to go with the couple he scored in the Community Shield. Even the Swansea fans joined in the applause when he was substituted and, on this evidence, the Dutchman looks set for another wonderful season.

Yet this was not one of those days when United were indebted solely to Van Persie. Moyes will probably be just as encouraged by the fact Danny Welbeck was responsible for the other two. Welbeck managed only one goal in the Premier League last season but has doubled that amount in a single match and, at the very least, it should soothe his confidence.

Maybe this was a good day, too, to remind Wayne Rooney that United is not such a bad place to be a footballer after all. If Rooney had feared there might be an abrasive reaction from the team's supporters, it was exactly the opposite when he entered the game as a 62nd-minute substitute. His name was sung loudly and he was involved in his side's last two goals, even if he did not celebrate either of them. Van Persie lashed in a beauty for the first, with a shot that was still rising as it arrowed into the roof of the net. Welbeck completed an emphatic victory with a lovely, clipped finish from Rooney's pass.

Altogether, that represents a day of huge satisfaction for Moyes and Wilfried Bony's goal was only a minor irritation given that Swansea were 3-0 down when his measured, right-foot finish found the bottom corner. Moyes's complaints about United's opening run of fixtures – even ringing the Premier League's chief executive, Richard Scudamore, to voice his suspicions – have sounded transparently like a man getting his excuses in early. As it turned out, his team have set themselves up perfectly for the fixtures against Chelsea, Liverpool and Manchester City. "A brilliant performance," Moyes described it, "against a Swansea team who, I believe, will do really well this season."

The names on the pitch are still the same, and the team are playing in the 4-2-3-1 system that Sir Alex Ferguson eventually came to favour. Giggs is still there at the age of 39 and Van Persie is still greedily accumulating goals from any distance or angle. So maybe we should not be surprised that it felt so much like business as usual. How many times, for example, have we seen this United team suddenly come into their own and ruthlessly punish the opposition as they did here with two goals in as many minutes? That was the period when the game lurched away from Swansea and, from then on, there was an air of inevitability.

Swansea had periods when they pinned back their opponents. Jonjo Shelvey, their recruit from Liverpool, was prominently involved and, at £5m, looks like he could be a smart piece of business. Michu was a difficult opponent for Rio Ferdinand and Nemanja Vidic and Bony, bar one wasteful miss, showed why Swansea have made him their record signing. Michael Laudrup was not exaggerating when he said his team had largely controlled the opening half an hour. "But then suddenly you saw the difference," he said.

This was clinical from Moyes's men. Van Persie's second was the pick of the lot but the first was not too shabby either, expertly adjusting his body shape as he read the bounce from Ryan Giggs's deflected pass, steadying himself with his first touch then hooking his shot past Michel Vorm.

Two minutes later, Patrice Evra aimed for Van Persie again. The ball skimmed off the striker's head into Antonio Valencia's path, running in from the right. His shot, on the half-volley, flashed across the six-yard area and Welbeck applied the final touch from point-blank range.

The pattern of the game continued in the second half: lots of Swansea possession but always the sense that United could add more goals. Van Persie had turned one shot wide and Evra had seen a header cleared off the line before the third goal arrived after 72 minutes. Rooney instigated the move, cutting in from the left. His decoy run was superb, and Van Persie let fly from almost 25 yards.

Soon afterwards, Moyes could be seen offering a polite, understated wave on request. The new era is off with a bang.