Manchester United 2-1 Everton
Man of the match Angel di Maria said his award should have gone to Manchester United goalkeeper David de Gea after his side recorded back-to-back wins for the first time under Louis van Gaal.
Di Maria put United ahead midway through the first half as he fired home from Juan Mata's cross for his third goal of the season.
But the Toffees had the chance to level matters at the break after Tony Hibbert was brought down by Luke Shaw on the stroke of half-time.
Leighton Baines stepped up to take the resulting penalty and aimed for the bottom left but was thwarted by De Gea, who made a big one-handed save.
Baines made up for his mistake 10 minutes into the second half as he picked out Steven Naismith with a pinpoint cross in the centre of the penalty box and the Scot headed home from close range with the equaliser.
The hosts were back in front just after the hour mark as Di Maria turned provider for fellow summer recruit Radamel Falcao, who slotted the ball past Tim Howard from six yards out to make it 2-1 with his first United goal.
De Gea did well to keep out a two late Leon Osman strikes, one from Gareth Barry and one fantastic effort from Bryan Oviedo as the hosts held on for the three points.
Di Maria told BT Sport 1: "I think David is the guy the who deserves this reward rather than myself he saved us throughout the game and particularly towards the end and the penalty as well so not only him but the rest of the side deserve this."
He also reserved praise for fellow newcomer Falcao and hopes the win is the start of things to come at Old Trafford, adding: "Yes, it's great for him.
"He's been looking for that first goal for a while now and he's it's very, very important for him now.
"I think not only him but ourselves are on the right road now and the main thing is we're fighting for victories and fighting for points.
"We've got the three points for the day and hopefully we can now move further up the table."
Red Devils boss Van Gaal was pleased with the three points that saw his side move up three places to fourth in the Barclays Premier League table.
He was full of praise for De Gea and Di Maria, and said: "When you hold the penalty out in the dying seconds you hold three magnificent saves, then you are great because you win the game because of that.
"When you give assists and also you score goals, what can you ask more of a player?
Meanwhile Everton boss Roberto Martinez was disappointed with the referee's decision not to stop the match when midfielder Steve Pienaar was injured, and United went on to score the winner.
On his return from a thigh injury which has sidelined him for the past month, Pienaar left the field after 64 minutes to another problem, two minutes after Falcao's strike which won the match for the home side.
Martinez, whose side travelled to Old Trafford after a long trip to southern Russia in the Europa League in midweek where they recorded a 1-1 draw at FK Krasnodar, said: "We're just disappointed with the second goal really.
"Steven Pienaar was on the ground, we made it clear to the referee to stop the game and he allowed it to play on and it was very, very disappointing.
"It should have never happened it was a game full of fair play, with both sets of players playing in the right manner and I thought the referee made a major, major error there."
Martinez, whose side did the double against United last season, rued his Eveeton's missed chances towards the end and added: "It was a very good game and a few chances appeared to open.
"I just felt that David de Gea made two world-class saves that probably stopped us getting a positive result. But even then, I thought that the time we were going forward we were creating and the second-half performance deserved a lot more than what we got."
