Moyes said: "We have a very good squad, we picked a side who have all played at the top level, they have all played in Europe this season, it was very much the midfield that played and won away to Alkmaar in the UEFA Cup.
"It wasn't a vastly weakened team at all, in fact I felt it was a strong team."
He added: "I would have expected that the team I put out would have been good enough to win.
"They are all disappointed, we all are. These things happen, unfortunately it has happened to us a few times.
"We missed lots of chances. We have scored a lot of goals at Goodison Park this season, so Oldham have done really well to come here and not concede.
"We have looked as if we could score lots of goals, but Oldham defended fantastically well and scored an opportunist goal just before the break.
"We didn't play with enough conviction and when you miss good early chances it becomes more and more difficult.
"We never really got to grips with them. We did not under-estimate them, we rated them very highly."
Latics boss John Sheridan admitted this was "the best result I have had as a manager" and added: "To come to a Premier League ground against a team who are really flying at the moment was a great performance.
"I told the players to really enjoy the occasion, because things like this might not happen to them again. I was so pleased and proud of them.
"I believed we could get something from the game, we have won seven on the trot away now, so we are doing something right.
"I can't praise them enough. The togetherness was outstanding, kids and experienced players."
Sheridan did not believe Everton put out a weakened side, adding: "I have watched Europe a few times, I have seen them in Europe, and I knew it was going to be a massive test for us whoever they put out.
"They have got some great kids like (James) Vaughan and (Victor) Anichebe, and whoever they fielded would have been tough for us.
"They have got such a strong squad. We knew we would be chasing the ball a lot but we did it."