Imperious Fellaini helps defeat United
The Belgian international, who hit the post in the first half with a close range shot and tormented United's defence throughout with his height, nodded in from a corner on 57 minutes.
The home side had enjoyed the better of the opening exchanges and, as the game developed, created the majority of chances in front of a noisy Goodison Park crowd.
United goalkeeper David de Gea had done well in the first half to keep the home side at bay by producing a string of first rate saves, most notably to deny Steven Pienaar's curling right-foot shot on the half-hour mark.
Wayne Rooney had a decent chance at the other end, only to direct his shot straight at Tim Howard.
However, with Fellaini working hard in the midfield, making his presence felt and linking up well with the Everton strikers, the home side always looked the more likely to break the deadlock.
The Belgian rose above Michael Carrick to get on the end of Leighton Baines' corner and power in his header into the corner of De Gea's goal.
Tom Cleverley's shot on 67 minutes was cleared off the line and the visitors then introduced new signing and league player of the 2011/12 season Robin van Persie, bought from rivals Arsenal last week. However, his debut did not affect the result.
With Phil Jagielka in fine form, clearing Cleverley's 67th-minute shot off the line, the Everton defence held firm, frustrating Manchester United and taking a deserved three points.
Jagielka told Sky Sports he was delighted with the result and hoped it could serve as a platform for the team to build on.
"We always knew it was going to be hard," the England defender said. "We scored a great goal and that gave us something to hang on to.
"It's fantastic to start a season with a win rather than the doom and gloom we usually start with.
"If we can build on this, get another point or three away at [Aston] Villa, then it will be a fantastic start for us."
"Manchester United are a great club, everybody was motivated," Fellaini said. "I tried to score and I think it's good for me"
Everton travel to Villa on Saturday afternoon, with Manchester United hosting Fulham.