Moyes spent 11 years as manager at Goodison Park before he replaced Alex Ferguson at United, but he was given a hostile reception from Everton fans and they rejoiced at full-time after an impressive 1-0 victory against their former boss.
And Roberto Martinez achieved what Moyes failed to during his tenure as the Merseyside club won at Old Trafford for the first time since 1992, ending the hosts 12-game unbeaten run.
Victory for Martinez's men stretched their unbeaten run to seven games and left United well off the pace in the race for the title.
Ross Barkley and Kevin Mirallas were restored to Everton's starting line-up at the expense of Leon Osman and on-loan winger Gerard Deulofeu, who impressed in his first Premier League start against Stoke City on Saturday.
Mirallas almost marked his recall with a goal after just eight minutes, when he cut in from the left and stung David de Gea's palms with a venomous right-foot drive from 20 yards out.
Howard had to be alert once again to thwart Shinji Kagawa and Giggs' header flashed wide of the far post as United got into their stride.
The Everton keeper was beaten after 28 minutes, but Rooney's deflected shot came back off the post.
Romelu Lukaku was a constant menace to the United defence in a frantic first half and De Gea saved the Belgium striker's effort with his feet five minutes before the break.
United were twice thwarted by the frame of the goal when Evra's header was tipped onto the crossbar by Howard and Danny Welbeck headed the rebound against the bar from eight yards out.
Mirallas' free-kick rattled the post for the visitors, but Everton won it late on when Lukaku's cross was tucked home at the back post by the onrushing Oviedo.
