An impressive performance from David Moyes' men saw them notch a win in their opening Premier League game with the Dutchman tasting defeat on his first appearance
Ahead of the game the big team news concerned the visitors with Robin van Persie named on the bench but Shinji Kagawa handed his Premier League debut. Elsewhere defensive problems were evident with Antonio Valencia at right-back and Michael Carrick partnering Nemanja Vidic in central defence.
Everton, meanwhile, also named one of their summer signings in their starting line-up and another on the bench. Steven Pienaar was given the nod in midfield, with Steven Naismith having to wait for his first Premier League start.
An entertaining first half at Goodison Park saw plenty of chances for both sides with the hosts enjoying the better opportunities and ensuring David de Gea had a busy evening in the United goal.
Marouane Fellaini was at the heart of all of Everton's best play and the Belgium international went close just after the 10 minute mark, controlling a throw from Leighton Baines, turning past Valencia and surging past Carrick but unable to beat De Gea who touched his effort onto the post.
Ten minutes later and Everton went close again. Some quick, one-touch passing proving too good for United and finding Nikica Jelavic inside the box and at a tight angle but again unable to beat De Gea.
From the ensuing corner, De Gea managed to punch clear but only as far as Pienaar on the edge of the box and his firm header was touched over again by the Spaniard.
The pressure from the home side was increasing and De Gea was once more called into action just after the half hour mark. A curling Baines effort forced the Spaniard to dive low, but he could only parry the ball which fell to Fellaini who fired the ball across goal and wide.
However, United were creating chances of their own and Rooney twice forced Tim Howard into saves. Firstly from a low free-kick that beat the wall and was heading for the bottom corner before the American tipped it wide and then with a snap shot after a good cross from Evra. Unfortunately for the England international it was straight at the former Manchester United goalkeeper who collected it gratefully.
As the half drew to a close Everton again went ramped up the pressure and almost managed to take a lead into the break. With five minutes left on the clock Leon Osman, on the turn, forced De Gea into another superb save before a Baines free-kick again saw the goalkeeper turning it away impressively at full stretch.
The second half began in much the same vein with the hosts rattling United’s crossbar within minutes of the restart. A cross from Tony Hibbert was knocked down by Fellaini to Osman who thundered a shot onto the crossbar.
With the woodwork still shuddering, United broke quickly downfield, working the ball swiftly to Danny Welbeck on the edge of the box, and about to shoot, until he was denied by a perfectly timed tackled from Jagielka.
However, just minutes shy of the hour mark, the goal Everton had been threatening all night finally came. A corner from the right found Fellaini, who out-jumped Carrick and thundered a header past De Gea, for a goal his performance had fully merited.
The visitors responded by sending Van Persie out to warm up, but minutes later Fellaini was tormenting the United defence again, heading straight at De Gea who was able to hold onto his effort.
Back came the visitors with the increasingly lively Kagawa teeing up Cleverley in the box, with Howard out of his goal, but his shot was hacked off the line by Jagielka.
That miss saw the introduction of Van Persie on 67 minutes, replacing Welbeck as the Red Devils chased an equaliser and he was swiftly followed by Ashley Young, on for the disappointing Nani.
With 10 minutes to go Van Persie slid in an enticing low cross for Kagawa, that had the United fans on their feet, but he was just beaten to the ball by Howard at his near post.
With time running out the visitors continued to press, inevitably, but Everton held firm, even allowing Fellaini to depart late on to a standing ovation, to pick up a deserved three points and leave Manchester United with plenty to ponder.