The Toffees battled back from a 4-2 deficit to claim a share of the spoils at Old Trafford.
Speaking after the match, Moyes insisted that his team were good value for the point that they achieved.
"The effort they put in was absolutely brilliant," Moyes told BBC Sport. "We came to win the game, we felt we had been a bit hard done by lately.
"Today was a chance to play in another big game and stop people thinking our season was over. I don't think at any time we deserved to be two goals down."
The draw has left Everton seventh in the Premier League table.