Van Persie came off the bench for the final 22 minutes at Goodison as United searched for an equaliser after Belgium midfielder Marouane Fellaini had headed Everton into a second-half lead.
But the Dutchman, signed from Arsenal for £24 million ($A36.34 million), was unable to make an immediate impact for his new club and Everton held on for a deserved win in their opening game of the season.
United goalkeeper David de Gea was the busiest player in the match, playing behind a second-string defence featuring only two recognised defenders: Nemanja Vidic, who had not played since December because of knee injury; and Patrice Evra, whose future has been a matter for debate all summer.
Winger Antonio Valencia was deployed at right-back while midfielder Michael Carrick was positioned alongside Vidic in the absence of the injured Rio Ferdinand, Phil Jones and Chris Smalling
De Gea made a series of superb saves to deny Steven Pienaar, Leighton Baines and Leon Osman in the first half, and Everton also hit the woodwork, through Fellaini and Osman, before they deservedly claimed the lead in the 57th minute.
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.
United were denied an equaliser when Phil Jagielka cleared Tom Cleverley's shot off the line, and there was no way back for them even after van Persie had replaced Danny Welbeck in the 68th minute.
"It was a great result," Everton manager David Moyes said.
"It's not very often you beat Manchester United, so we will enjoy it.
"There was a lot of attention on United before the game, but I think a lot of people forgot that Everton are a good team as well.
"We had some really good chances and their goalkeeper made some good saves.
"Fellaini was excellent.
"We know he is capable of that.
"He caused them some problems.''
Fellaini said: "It was a good goal, it was a good cross, a good corner. It was good for me and for Everton.
"Everton had a lot of chances in the first half and we had to keep going and hoped the team scored.
"I think everybody was motivated. Manchester United is a great club. Everybody is motivated, and I think today everybody played well."
Everton defender Phil Jagielka said the Toffees had identified Carrick's selection in United's defence as a weakness to be targeted.
"He's not a natural central defender," Jagielka said of Carrick.
included, and told them never to forget what had just happened.
