The Scot, who enjoyed 11 successful years at Goodison Park before replacing Alex Ferguson at Old Trafford at the end of last season, was booed by the home fans when he appeared from the tunnel at the start of the game.
His fortunes did not improve as Everton took command with a Leighton Baines penalty in the 28th minute following a Phil Jones handball, before the hosts doubled their lead through Kevin Mirallas’s low finish just before the break.
Despite controlling possession, a toothless United struggled to carve out any meaningful chances and rarely troubled Everton keeper Tim Howard before falling to their 11th Premier League defeat of the season.
“We had a lot the ball, passing, and controlled the majority of the game,” said Moyes, whose side are seventh in the table with 57 points from 34 games.
“But we didn’t have a cutting edge and got done on the counter attack. We gave away two dreadful goals.”
United fielded the attacking quartet of Shinji Kagawa, Juan Mata, Nani and Wayne Rooney, but lacked a creative spark when they got into Everton territory.
Moyes, however, said the wheels were in motion to strengthen the squad for next season.
“We couldn’t find the telling pass, or clever play in the final third that makes the difference,” he said.
“I thought we did well, were the better team at 2-0 down in the first half.
“We’ll do everything we can to win the remaining four games. We are under way with what we are doing to improve next season and try to give ourselves a better chance to compete at the top end of the table.”
Martinez again showcased his managerial talents to mastermind a seventh victory in eight matches which kept them one point behind fourth-placed Arsenal with three games to play.
The Spaniard, who led Wigan Athletic to FA Cup success last season, has been a revelation since taking over from Moyes and his team have played an attractive and positive style of football which has earned many plaudits.
Everton enjoyed just 38 percent possession against United, their lowest of the season, but were rarely troubled by the visitors and Martinez praised their “electrifying” counter-attacking performance.
“Sometimes you need to adapt to what you face on the pitch,” he told the BBC.
“It was a case of the quality United had on the pitch, we had to be solid, organised and use the dynamic football we have.
“The 2-0 lead allowed us to be more economical with our performance. We were very clear in what we had to do and we did that. We were electrifying on the counter.”
(Reporting By Tom Hayward; editing by Toby Davis)