Bryan Oviedo's 86th-minute goal secured Everton's 1-0 victory as Moyes's difficult start continued at champions United, who slipped 12 points behind Premier League leaders Arsenal after 14 matches of the season.
"I'm very disappointed," Moyes told the BBC after United's first defeat in 13 games.
"We needed a bit of good fortune and we didn't get it. Everton kept at it and showed us what a good team they are.
"We didn't defend the goal well enough so it was our own doing. We had some chances as well as did Everton. They have been playing well and that's why they are up near the top of the league.
"It wasn't about me tonight, I was fortunate to manage Everton and I'm fortunate to be managing Manchester United. It's a big catch-up for us now."
Moyes failed to win at Old Trafford in his 11-year spell in charge of Everton and his new club dropped to ninth in the table ahead of Saturday's home game against Newcastle United.
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