Van Persie opened the scoring after 34 minutes, controlling a bouncing ball in the penalty area and swivelling expertly to hook it in right-footed from eight metres.
Welbeck doubled the lead two minutes later with a close range tap-in and Van Persie added a third after 72 minutes with a fine individual effort.
Wilfried Bony pulled one back for Swansea but Welbeck dinked in his second goal in stoppage time following a pass by substitute Wayne Rooney to give the champions a comfortable victory to start their title defence.
Image: Robin van Persie
Photograph: Michael Steele/Getty Images
Pressure on Moyes as City and Chelsea up the ante
EPL: Moyes driven by fear factor at Manchester United
'I believe in this team, we are the champions'
Schedule: English Premier League 2013-14
It's Mourinho 1, Moyes 0 in parallel mindgame League