However, the Toffees almost snatched the unlikeliest of winners when they broke from a United corner to pour forward in numbers.
Atkinson blew for full-time before Phil Jagielka had drilled his shot at Edwin van der Sar and Moyes was disappointed the official did not allow play to continue.
"I think the two goals were scored after 90 minutes and he put three minutes of extra-time," Moyes told Sky Sports.
"After about three-and-a-half minutes, he allowed Manchester United to take a corner kick.
"We broke that fast I don't think it could have taken more than eight, nine, 10 seconds before we played the final pass and he blew up, so that was wrong."
Moyes, though, refused to allow this to take the gloss off a stirring comeback at Goodison Park.
He added: "It doesn't take anything off. The players' character and desire was shown to get back into the game.
"We rode our luck a little bit and they had one or two chances, but we had a great chance at 3-1, but Leon Osman didn't take it and I thought it was another one of those days where we were not going to score."