Steven Gerrard scored two penalties and hit the post with a third - for which Nemanja Vidic was sent off - before Luis Suarez finally put United out of their misery.
Moyes had issues with some of the decisions made by referee Mark Clattenburg, not least Liverpool's third penalty and Glen Johnson's handball which went unpunished, but had the good grace to accept his side were out-played - something increasingly disgruntled United fans will not like hearing.
"We didn't play well, Liverpool deserved their victory," said the Scot after a ninth league defeat of the season and a fifth at home.
"Some decisions were correct, some were incorrect. but overall I'm disappointed.
"It is difficult to explain it."
This season United have won just once against clubs in the top nine and Moyes said that was obviously something that had to change.
"That tells you we're not doing as well as we should be. We have to play better and make ourselves harder to beat, harder to play against and we are also going to have to make sure we are creating and taking more opportunities," he said.
"We have a lot to do to do both those things. I think the job was always going to be hard. Yes, it is harder (than I thought) I would say so."
Following Ferguson was never going to be easy but the way Moyes is conducting himself is almost the opposite of his predecessor.
Counterpart Brendan Rodgers was taken aback the Red Devils boss played the underdog card in his pre-match press conference.
"I was probably surprised when I heard we were supposedly coming to Old Trafford here as favourites," said the Liverpool manager.
"I would never say that at Liverpool - even if I was bottom of the league."
Moyes tried to justify his pronouncement by saying: "I just thought Liverpool were above us in the league and playing well and I thought any average person would have said the same thing."
The United boss felt Liverpool's second - just a minute after the break - changed the complexion of the game.
"I thought the real turning moment was the second penalty just after half-time," he added.
"To lose that made it dificult for us. There wasn't a great deal in it at half-time.
"There was a handball from Glen Johsnon in a similar position (to where Rafael was penalised for the opening spot-kick) which wasn't given. Nevertheless I can't have any qualms about it.
"There were quite a few decisions but I won't speak about it, it seems the best way to go about it. Us managers are better not commenting on decisions in games.
"I felt the players looked in good shape going into it. We just didn't quite get to the standards required to beat Liverpool today.
"I've said it all week I've seen confident and well-motivated and hard-working players.
"I think goals change games very quickly. We didn't start as well as we liked but I thought we started to grow into the game and the penalty kick changed that in the first half."