Steven Gerrard scored a penalty in each half, and struck the post with another spot-kick, while Luis Suarez added a late third goal to wrap up a comprehensive win for Liverpool.
The visitors' first penalty was awarded for a handball by Rafael, the second a minute after half-time when Phil Jones nudged over Joe Allen and the third when Daniel Sturridge went down under a challenge from Nemanja Vidic.
The United captain was shown a second yellow card for the tackle, although replays suggested Sturridge dived, while Moyes' side had two penalty appeals waved away with the score 2-0.
Moyes told Sky Sports: "It's hard to assess the game which was littered with penalty kicks and decisions, some were right and some were wrong.
"It was a tough day for us. At half-time, we were 1-0 down to a penalty kick, but I didn't think there was a lot in it.
"But to give away one 20 seconds into the second half made it a real uphill climb after that."
When asked if Liverpool were the better side, Moyes replied: "Yeah. They played better than us.
"Some decisions went for them which were important at the right time with the penalty kicks they got. They earned them at times, and some occasions they didn't earn them at all.
"It's better if I don't talk about them, that's up to the authorities and you hope they make the right decisions."
Moyes was also asked if the players played for him and the United boss replied: "I think they tried to do it, but I thought Liverpool played well.
"The first half I didn't think there was a great deal in it, but the penalty kick was the turning point in the first half and gave them something. To lose a goal right after half-time was tough."
United trail local rivals Manchester City in fourth place by 12 points and Moyes will not concede defeat in the quest to finish in the top four.
He added: "We'll fight to the end and wait and see how we get on. It will be tough, but we'll see what we can do."