First-half goals from Leighton Baines and Kevin Mirallas were enough to earn Everton, who won 1-0 at Old Trafford in December, a first double over United since 1969/70 and humiliate their former manager.
It was a 10th loss since the turn of the year, which means Champions League football is now beyond United, champions by 11 points last year.
But even though his team managed just two shots on target and appeared utterly directionless in possession, Moyes maintained his team were unfortunate and insists supporters will stick with him.
He said: “We couldn’t find the telling pass, or clever play in the final third that makes the difference. I thought we did well, were the better team at 2-0 down in the first half.
“We had good play, we had good control of the game, we had the bulk of the ball. I thought the two goals we conceded were rank, rotten.
“The supporters have been incredibly behind the team and supported the team throughout. They understand it has not been good, I recognise it has not been good, it needs to be better.”
It was an assessment that had former Liverpool defender Jamie Carragher claiming he worried for Moyes.
Republic of Ireland assistant manager Roy Keane must have been shocked once again at the decline of his old club but the impressive displays of Seamus Coleman and James McCarthy made his trip to Goodison Park more than worthwhile. McCarthy was utterly dominant in the middle of the field.
Everton were dominant from the start with a staggering contrast in the energy, movement and intelligence between the three-man withdrawn forward lines of Steven Naismith, Mirallas and Ross Barkley and United’s Nani, Shinji Kagawa and Juan Mata.
The home side’s trio tracked back and caused far more problems than United’s three, who allowed Everton oceans of space down the flanks and it was hard to believe the hosts had last played on Wednesday, while United had been preparing for a week and a half.
Coleman was first to exploit the space when his cross for Mirallas was glanced away by Chris Smalling and Barkley drove over from the edge of the area.
Everton took the lead when Phil Jones slipped and threw an arm out to block Lukaku’s shot and Baines drove in from the spot just before the half hour.
The hosts were happy to soak up United’s aimless possession and punished them on the break once again. Mata gave up possession in midfield and Naismith switched to the right.
Coleman guided the ball through a gap for Mirallas, who scored Everton’s second with a firm cross-shot after being played onside by Alex Buttner.
After Kagawa could not stretch Tim Howard beyond a routine save at the start of the second half, Moyes tried a 4-3-3, bringing on Antonio Valencia and Javier Hernandez but they remained toothless.
Darren Fletcher sliced wide for United but Naismith should have score a third at the other end, the Scot missing from a good position after being picked out by Mirallas’ cross.
Naismith went closer when he was fed by Coleman and had a curling shot pushed past the post by David de Gea after a quick break. Rooney had an effort palmed away by Howard before Smalling wasted a decent headed chance in added time.
Moyes refused to contemplate the prospect of Danny Welbeck being another United player looking to move on this summer, while Everton manager Roberto Martinez was close to running out of adjectives for his team.
“We had the better chances and they had the ball,” Martinez said. “We were happy for them to have the ball and used it to our advantage. We were full of energy, dynamic and electrifying on the counter-attack.”
EVERTON (4-2-3-1): Howard 7; Coleman 8, Stones 7, Distin 7 (Alcaraz 46, 7), Baines 7; McCarthy 8, Barry 7; Mirallas 8 (McGeady 90), Naismith 7, Barkley 6 (Osman 70, 6); Lukaku 6.
MANCHESTER UNITED (4-2-3-1): De Gea 6; Smalling 6, Evans 6 (Hernandez 61, 6), Jones 5, Buttner 5; Fletcher 7, Carrick 6; Nani 3 (Valencia 61, 6), Mata 6, Kagawa 5 (Welbeck 75, 6); Rooney 6.
Referee: Mark Clattenburg.