Leighton Baines and Kevin Mirallas got the goals on Merseyside to earn Everton - who prevailed by a one-goal margin at Old Trafford back in December - a first six-point haul against United since the 1969/70 campaign.
Victory on Sunday also keeps Roberto Martinez's side in the hunt for UEFA Champions League qualification - as they sit one point adrift of fourth-placed Arsenal - while United remain six back on Tottenham Hotspur in the battle for sixth and a guaranteed Europa League berth.
Everton ensured that it would be another painful afternoon for Moyes - and former favourites Wayne Rooney and Marouane Fellaini - as they once again displayed the kind of passion and determination which has carried them to eight successes in their last nine outings.
They had already seen one penalty appeal waved away by referee Mark Clattenburg and Steven Naismith fire over the crossbar when well placed by the time Phil Jones slipped inside the box and handled a Romelu Lukaku shot shortly before the half-hour mark.
Baines made no mistake from the spot as he preserved his perfect conversion record in Premier League competition by drilling down the middle as David de Gea dived away to his right.
That lead was doubled two minutes before half-time as Kevin Mirallas sprung the offside trap and crashed home from a tight angle.
The result and performance from both teams showed just how far United have regressed and how much Everton have come on under Martinez.
For United, Champions League qualification is now mathematically impossible. For Everton, their dreams of dining at Europe's top table are very much alive and few would back against them based on this performance.
As expected, Moyes emerged from the tunnel to a loud chorus of boos from the home fans. He stood on the touchline, seemingly unaffected by the negative reaction from the fans who cheered him on for 11 years.
On the pitch it was Everton who were taunting their opponents. They dominated possession and United could only chase shadows as Everton flicked the ball across the well-manicured turf with ease.
Rooney, his every touch booed, dropped too deep, as did Juan Mata, leaving United without a target man up top.
Around the 15-minute mark, Everton had their first real chance. Seamus Coleman found Naismith in the box but his shot was blocked by Evans' right elbow. Clattenburg waved away appeals for a penalty.
Moments later the referee had no choice but to point to the spot when Jones inexplicably stuck out his right arm to stop Lukaku's shot.
Baines blasted the ball down the middle inches away from De Gea's out-stretched leg.
The seemingly inevitable second came just before the break when the excellent Coleman played in Mirallas - played onside by Alexander Buttner - who beat De Gea from close range.
Tim Howard was finally called into action in first-half injury-time when Evans headed into the American's hands.
Rooney had a much better chance after the restart, but he dallied, allowing substitute Antolin Alcaraz to come to the rescue.
Moyes brought on Antonio Valencia, Javier Hernandez, and later Danny Welbeck but nothing changed on the pitch, as Everton remained in control.
Tempers then flared when James McCarthy saw yellow for a rash challenge on Mata and Rooney reacted with a late two-footed tackle on the Irishman.
Naismith wasted two good chances to make it 3-0 while Howard denied Rooney a goal at the other end.
There was no way through and Moyes endured taunts of "sacked in the morning" shortly before he disappeared down the tunnel at the final whistle.