Luis Suarez looked to have grabbed a winner deep into injury time when he scored from Sebastian Coates's knockdown, but the referee's assistant controversially ruled the effort out for offside.
Moyes accepts the officials got the decision wrong, but feels a free-kick should not have been awarded in the first place for Leon Osman's challenge on Steven Gerrard.
"I think we were a little fortunate with the goal," Moyes told Sky Sports. "The goal is not offside, but I don't think it was a foul on Gerrard by Osman in the middle of the pitch, I think the referee got that wrong.
"But overall I think the referee had a pretty steady day considering the atmosphere and what goes on here."
Moyes was delighted with the battling spirit of his side as they came from 2-0 down to level matters before half-time thanks to goals from Leon Osman and Steven Naismith.
"It was a good game, a really exciting game," added Moyes. "It was a disappointing start for us, but the boys showed great character to come from 2-0 down in a derby and get back into it and I thought we were a little bit unfortunate not to go in 3-2 up at half-time."
Moyes had no complaints about Suarez's celebration when the striker dived in front of him in reference to the Scot's accusation in the build-up to the match that the Uruguayan went to ground too easily.
"I thought it was great. I would probably have done the same if I had been the player, but he is going to have to dive in front of a lot of managers then I think.
"I might have done the same to him, if we had won 3-2 I might have done one in front of him as well."
Moyes revealed he also had a word with captain Phil Neville at half-time after he was booked for diving in the first half.
"Well I have got to say Phil Neville went down too easy so I have told him about it," continued Moyes.
"I think if you go down too easy and you are a player then the manager should tell them as you don't want to get a reputation for being that sort of player."