Marouane Fellaini capped an impressive performance by heading in a corner from former United midfielder Darron Gibson in the 57th minute.
"I don't think anybody tonight was considering Everton at all," Everton manager David Moyes told Sky Sports television. "A lot of people forgot Everton aren't a bad team, and Goodison is a difficult place to come and win."
Tom Cleverley came closest to levelling for last season's runners up, but his shot was cleared off the line by Everton defender Phil Jagielka.
Signing Van Persie from Arsenal last week was meant to provide the additional firepower the 19-time champions need to regain the Premier League title from rivals Manchester City.
But the Netherlands striker, who cost a reported 24 million pounds ($38 million), failed to even hit the target after coming off the bench in the 68th.
It was the first time Alex Ferguson's side has lost a league opener since 2004.
Injuries, though, meant there were just two recognized defenders in United's back four, with winger Antonio Valencia deployed at right back, and midfielder Michael Carrick alongside Nemanja Vidic in the centre.
And a string of fine saves by United goalkeeper David de Gea prevented Everton from being in front by half time.
Fellaini, playing just behind striker Nikica Jelavic, was the dominant force in United's penalty area.
The Belgium midfielder had already headed over Leighton Baines' corner before he collected a throw-in and turned into the penalty area, holding off Carrick, to hit the post from a tight angle.
He turned creator, chesting the ball back for Steven Pienaar to slice a shot wide, and then knocked down a cross that Leon Osman hit on the turn only for De Gea to bat away the shot.
De Gea also denied Jelavic, tipped over a long-range header from Pienaar, and prevented Baines' deflected free kick sneaking into the top corner.
United striker Wayne Rooney had a quiet night as he marked 10 years since making his senior debut for Everton, hitting two shots too close to goalkeeper Tim Howard, and seeing a late first-half header punched clear.
After the break, Fellaini's header across goal was smashed against the crossbar by Leon Osman but his contribution was rewarded with a goal in the 57th.
"Fortunately, the big man (Fellaini) put his head where it hurts and we had something to hang on to," Jagielka said.
Van Persie's arrival meant Rooney shifted to the left, but it was another new signing, Shinji Kagawa, who made a bigger impression. The Japan playmaker forced Howard to race off his line to block his run and shot.
And Everton held on to record just its fourth home win against United in 21 Premier League games.
"It's nice to send the Everton fans on the first day of our season home with a win and not the doom and gloom we normally start with," Jagielka said.