First-half goals by Leighton Baines - from the penalty spot - and Kevin Mirallas kept Everton one point behind fourth-place Arsenal with three games left, and guaranteed United won't be playing in Europe's top competition next season.
Languishing in seventh position, six points behind sixth-place Tottenham, United may not even qualify for the Europa League - an indication of the deposed champions' decline since winning the Premier League last year by 11 points.
Moyes left Everton after 11 years to take charge of United. But while his first season at Old Trafford has been extremely disappointing, Moyes' replacement at Everton - Roberto Martinez - has been widely praised for guiding the Merseyside team to its best haul of points in Premier League history already.
Everton, which has achieved the double over United for the first time since the 1969-70 season, played much the better football here and still has a great chance to qualify for the Champions League's qualification round - something Moyes achieved only once.
A flurry of boos greeted Moyes as he emerged from United's team bus before the game and then from the tunnel prior to kickoff, with many home fans unhappy at the circumstances surrounding his departure to Old Trafford and also his attempts at poaching Leighton Baines and Marouane Fellaini from Everton over the summer.
Fellaini ended up moving for 27.5 million pounds ($43 million) on deadline day in September - he was an unused substitute here - but Baines stayed at Everton and showed Moyes just what he is missing by having an excellent game and driving home a 28th-minute penalty to open the scoring after Phil Jones handled Romelu Lukaku's shot.
With Mirallas running beyond a wide-open defense to add a second from Seamus Coleman's pass in the 43rd minute, it amounted to an uncomfortable first 45 minutes for Moyes, who was sat near a spectator dressed as the Grim Reaper and waving an imitation scythe in Moyes' direction. The spectator was escorted out of the stands midway through the first half.
The margin of defeat could have been a whole lot bigger for United, whose left side of fullback Alex Buttner and midfielder Shinji Kagawa in particular was over-run by the marauding Mirallas and Coleman.
Steven Naismith wasted great chances in either half and David de Gea pushed aside a curler from Lukaku.
Everton goalkeeper Tim Howard, on the other hand, was called into action just twice, saving from Jonny Evans and Wayne Rooney. The frustrations of Rooney, United's captain for the day against his boyhood club, were highlighted by a reckless challenge on Everton's James McCarthy that was fortunate to escape a booking.