New boss David Moyes has had a difficult summer, failing to land his transfer targets and having to answer numerous questions about Wayne Rooney's future.
The England striker was a second-half substitute but he was thoroughly overshadowed by Van Persie, who volleyed United ahead in the 34th minute and scored a sumptuous third 18 minutes from time.
Danny Welbeck netted a second in between, while there was consolation for Swansea in the form of a debut goal for Wilfried Bony (82) before Welbeck had the final say with a deft chip in injury time.
It was a miserable first afternoon of the Barclays Premier League season for Arsenal boss Arsene Wenger as Aston Villa came from behind to win at the Emirates Stadium.
The Gunners have not made any major signings yet this summer and the fans were far from happy as two goals from Christian Benteke put the visitors on their way to a 3-1 victory.
Arsenal were ahead after only six minutes through Olivier Giroud, but Wojciech Szczesny gave away a penalty and, although the keeper saved Benteke's spot-kick, the Belgian headed in the rebound in the 22nd minute.
And 16 minutes after half-time Benteke had another chance from the spot when Laurent Koscielny fouled Gabriel Agbonlahor, and this time he only needed one chance.
It got even worse for Arsenal six minutes later when Koscielny was shown a second yellow card, and Villa made sure five minutes from time as Antonio Luna scored on his debut.
New Everton boss Roberto Martinez had to settle for a point from his first competitive game in charge as the Toffees drew 2-2 with Norwich at Carrow Road.
After a goalless first half, Steven Whittaker put Norwich in front in the 51st minute but teenager Ross Barkley drilled in his first goal for Everton 10 minutes later and in the 65th minute the visitors took the lead through Seamus Coleman.
Their lead lasted only six minutes, though, as Ricky van Wolfswinkel grabbed a debut goal for Norwich.
There was no dream start to life in the Premier League for Cardiff, who were beaten 2-0 at West Ham.
Joe Cole netted on the turn in the 13th minute and Kevin Nolan made the points safe for the Hammers 14 minutes from time.
Liverpool were the first winners of the new season as they saw off Stoke 1-0 in the early game at Anfield.
Daniel Sturridge scored the only goal in the 37th minute with a precision shot from the edge of the area.
Stoke goalkeeper Asmir Begovic was the man of the match as he kept Liverpool at bay with a host of fine saves, and it looked like Potters boss Mark Hughes would get a point in his first competitive game when the visitors were awarded a late penalty for a handball by Daniel Agger.
There was only a minute of normal time left when Jonathan Walters stepped up but debutant goalkeeper Simon Mignolet became an instant hero as he denied the Irishman and then Kenwyne Jones from the rebound.
England hero Rickie Lambert's great week continued as he netted a 90th-minute penalty to give Southampton a 1-0 victory over West Brom at The Hawthorns.
It was the same score at the Stadium of Light where Patjim Kasami's first goal for Fulham earned the Cottagers victory over Sunderland.
Saturday's Premier League Results
Liverpool 1-0 Stoke
Arsenal 1-3 Aston Villa
Norwich 2-2 Everton
Sunderland 0-1 Fulham
West Brom 0-1 Southampton
West Ham 2-0 Cardiff
Swansea 1-4 Manchester United
Sunday's Premier League Fixtures
Crystal Palace vs Tottenham
Chelsea vs Hull
Monday's Premier League Fixture
Manchester City vs Newcastle