The Scot spent 11 years at the Merseyside club before joining United at the start of the season and was jeered as he emerged from the tunnel at the start of the game.
Moyes' mood will not have been improved by United's disjointed display, as first-half goals from Leighton Baines and Kevin Mirallas kept Everton's UEFA Champions League hopes alive.
The pressure was on Roberto Martinez's men to respond after Arsenal's 3-0 win at Hull City earlier in the day and Everton did just that with a dominant performance.
Baines - often linked with a move to Old Trafford in recent times - put Everton in front with a penalty after Phil Jones had handled Romelu Lukaku's shot midway through the first half.
Shortly before the break, Mirallas gave the hosts some breathing space with a fine low finish across David de Gea after getting on the end of Seamus Coleman's pass.
The result keeps Everton a point behind Arsenal in the race for fourth, while United remain seventh, six points adrift of Tottenham in sixth.
Steven Naismith, recalled to the side along with James McCarthy, ought to have given Everton the lead in the 22nd minute.
Lukaku's knockdown found Naismith all alone inside the area, but the Scotland international blazed wildly over the crossbar from just inside the area.
That miss was rendered irrelevant six minutes later, though, as Baines put the hosts in front from the penalty spot.
Jones, having lost his footing, stuck out an arm to block Lukaku's shot and, after referee Mark Clattenburg awarded the spot-kick, Baines sent his effort straight down the middle.
Everton doubled their lead two minutes before the break, Mirallas finishing from a tight angle after being slipped in by Coleman.
Lukaku had the chance to make the game safe in the 62nd minute, but the Belgian fired straight at De Gea after being picked out by a sensational pass from Mirallas on the halfway line.
Naismith spurned another opportunity before being denied by De Gea at full stretch 10 minutes from time.
Former United goalkeeper Tim Howard, who this week signed a contract extension, was not seriously called into action until three minutes from the end, when he made a superb save from ex-Everton man Wayne Rooney to preserve his clean sheet.
The win, which secured Everton's first league double over United for 44 years, provided a welcome tonic for Martinez and his side after the disappointment of Wednesday's surprise 3-2 defeat to Crystal Palace, while the result leaves Moyes with plenty to ponder.
