Bryan Oviedo scored the only goal of the game four minutes from time as Roberto Martinez succeeded where Moyes had failed by masterminding the Merseyside club's first win at Old Trafford for 21 years.
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It was a miserable game for Moyes, who was given a rough ride by Everton fans in the first meeting between the two clubs since the Scotsman left Goodison Park to replace Sir Alex Ferguson at United.
An unmarked Oviedo popped up at the back post to end United's 12-match unbeaten run in all competitions with a defeat that leaves them down in ninth place, 12 points behind leaders Arsenal.
Moyes praised Everton - who are five points better off than United in fifth place - and concedes his new charges have made life difficult for themselves in the race for the Premier League crown.
He told Sky Sports News: "I'm very disappointed, but that's the way it goes. It was tight and we missed chances, but Everton took theirs.
"We needed a bit of good fortune and we didn't get it. We hit the post a couple of times and Tim Howard made some saves. Everton kept at it and showed what a good team they are.
"We didn't defend the goal well enough so it was our own doing. We had some chances, as did Everton. They have been playing well and that's why they are up near the top of the league."
When asked about his reception from the Everton fans, Moyes added: "It wasn't about me tonight.
"I was fortunate to manage Everton and I'm fortunate to be managing Manchester United. It's a big catch up for us now."