Van Persie came off the bench for the last 22 minutes as United searched for an equaliser after Belgian midfielder Fellaini headed Everton into a second half lead.
But the Dutch forward, signed from Arsenal for £24 million last week, was unable to make an immediate impact for his new club and Everton held on for a well deserved win in their opening Premier League game of the season.
United goalkeeper David de Gea endured some difficult moments last season, but the young Spaniard made a series of superb saves to deny Steven Pienaar, Leighton Baines and Leon Osman in the first half.
David Moyes’s side also hit the woodwork through Fellaini and Osman, but Everton’s pressure finally paid off in the 57th minute.
With Rio Ferdinand ruled out with a groin injury suffered at the weekend and fellow defenders Phil Jones and Chris Smalling already sidelined, United boss Sir Alex Ferguson was forced to play midfielder Michael Carrick out of position in central defence.
And Fellaini took full advantage of Carrick’s lack of aerial power as he rose above the United star to thump a close-range header past de Gea from Darron Gibson’s corner.
Ferguson’s team were denied an equaliser when Phil Jagielka cleared Tom Cleverley’s shot off the line and, although van Persie replaced Danny Welbeck in the 68th minute, there was no way back for United.