Moyes was greeted with a smattering of boos as he walked out of the tunnel before the match, but it was the loss, which definitively ended United’s hopes of qualifying for the Champions League, that will trouble him more.
Before Liverpool’s 3-2 win at Norwich City earlier in the day, United had boasted the best away record in the division, but they were swept aside by Roberto Martinez’s effervescent Everton team.
Leighton Baines, a United transfer target until he signed a new long-term deal earlier this year, opened the scoring with a 28th-minute penalty and Kevin Mirallas added a second goal shortly before half-time.
A United comeback never looked likely as Moyes’s beleaguered side slipped to their 10th defeat of 2014 in all competitions and their 11th loss of the league campaign.
But Everton, who won 1-0 at Old Trafford in December, remain a point behind fourth-place Arsenal with three matches remaining after only a third win over United in 19 league matches at Goodison Park.
Everton played with the greater incision from the start, with Seamus Coleman bounding forward from right-back at every opportunity.
An early cross towards Mirallas was glanced away by Chris Smalling and Ross Barkley lifted a shot over from the rebound.
Phil Jones then did well to block efforts from Romelu Lukaku and Coleman as Everton began to dominate.
The hosts had a penalty appeal turned down when Jonny Evans threw himself at Steven Naismith’s effort, before the Everton player wasted a fine opportunity when he thumped over from Lukaku’s flick.
With Juan Mata, Shinji Kagawa and Nani — making his first start since December 7 — playing behind Wayne Rooney, United were listless as an attacking force.
From a spell of attacking possession for the visitors, Everton broke quickly and after Jones slipped, he stuck out an arm to block Lukaku’s curling shot, obliging referee Mark Clattenburg to award a penalty.
– Everton fans goad Moyes –
Baines guided the spot-kick down the middle for his sixth goal of the season.
After that, Everton soaked up more aimless play from United and doubled their advantage just before the interval.
Baines won the ball from Mata and Naismith knocked the ball out to the right for Coleman.
The Irishman slipped a pass through for Mirallas, who was played onside by Alex Buttner, and the Belgian placed a shot inside the far post.
Tim Howard was finally worked in first-half added time when a Rooney corner hit Sylvain Distin and the Everton goalkeeper was forced into a save.
Everton lost Distin to injury at the break and the visitors immediately showed more threat, although substitute Antolin Alcaraz did well to deny Rooney a shooting chance after the United captain had turned away from John Stones inside the area.
When Everton failed to deal with a Rooney free-kick from the right, Kagawa’s low drive was held at the second attempt by Howard.
Alcaraz might have done better when he could only make weak contact with a free header from a Mirallas corner.
After Kagawa headed straight at Howard from Buttner’s deep cross, Moyes tried to shake his team up by bringing on Antonio Valancia at right-back and also throwing on striker Javier Hernandez.
Darren Fletcher sliced wide for United, but Naismith should have scored a third at the other end, the Scot missing from a good position after being picked out by Mirallas.
Naismith went closer when he was fed by Coleman and had a curling shot pushed past the post by United goalkeeper David de Gea after a quick break.
Rooney had an effort palmed away by Howard in the closing stages, but United were well beaten and Everton supporters chanted ‘You’re getting sacked in the morning!’ at Moyes as the match drew to a close.