Robin van Persie had a Manchester United debut to forget as Everton's Marouane Fellaini stole the show at Goodison Park.
Fellaini capped a superb individual performance with the only goal of the game in the second half.
Van Persie, United's new £24million signing, came off the bench but, like the rest of his team-mates, could not cope with David Moyes' men.
Jump for joy: Marouane Fellaini gets up to head the ball past United keeper David De Gea
Wayne Rooney had a couple of chances for the visitors but it was the home side, with Fellaini making them tick, who bossed the opening 45 minutes.
Osman then smashed an effort off the crossbar shortly after the restart and Everton kept up the pressure until Fellaini powered home a header just short of the hour mark.
Van Persie left the bench to warm up just after the goal and was on the pitch in place of Danny Welbeck 10 minutes later.
But Everton kept him quiet and deserved the three points.
Full report to follow...
And we're off: Fellaini sprints away to celebrate after scoring for Everton
Fan-tastic: Marouane Fellaini celebrates his goal with Nikica Jelavic in front of the Everton supporters
No fear: Manchester United goalkeeper David De Gea clears under pressure
No holding back: Everton left-back Leighton Baines and United winger Nani get stuck in
Welcome to the Premier League: Shinji Kagawa was named in the Manchester United starting XI
On the ball: Robin van Persie warmed up before taking his place on the Manchester United bench
Game for a laugh: Van Persie shares a joke with new team-mate Anderson as he watches the first half
Worth a try: Leon Osman has a go as United skipper Nemanja Vidic keeps a close eye on him
Going it alone: Marouane Fellaini hit the post from a tight angle early in the first half
Experience: Veteran midfielder Paul Scholes tries to get Manchester United going
Hard to handle: Everton striker Nikica Jelavic tries to get away from Patrice Evra (left) and Tom Cleverley
Another new recruit: Kevin Mirallas is introduced to the Everton supporters before the kick-off
Star man: Marouane Fellaini caused Manchester United plenty of problems