Byran Oviedo's 85th-minute strike was enough to condemn Moyes's side to a defeat which leaves them 12 points off Premier League leaders Arsenal.
"Yes, we have a big catch up," Moyes told Sky Sports News.
"We're very disappointed that we never took a couple of chances. Tim Howard made one or two saves, but Everton are good side and they showed that throughout.
"We defended well at times but Everton played well, and that is why they're towards the top of the league."
United are now ninth in the division.