Moyes queries referee decisions
Steven Gerrard scored twice from the spot at Old Trafford before missing the chance to complete a hat-trick when hitting the post with another penalty 12 minutes from time.
United saw Nemanja Vidic sent off for the foul that led to Liverpool's third penalty, and went on to concede again as Luis Suarez found the net in the 84th minute.
"It was a game littered with penalty kicks and decisions - some were right and some were wrong," Moyes told Sky Sports.
"To give away one (a penalty) 20 seconds into the second half made it an upward climb.
"They (Liverpool) played better than us. Some decisions went for them and on some occasions they earned it but on others they didn't.
"I think that's just the way things are; it's best we don't talk about it."
Moyes was particularly aggrieved by United being denied a penalty when Glen Johnson appeared to handle in the area.
The United boss continued: "I thought it hits his hand, it's not a lot different from (the penalty given against) Rafael.
"I think Liverpool played well but in the first half there wasn't a great deal in it."
Despite suffering another setback in a disappointing first season in charge at Old Trafford, Moyes has refused to concede defeat in the chase for UEFA Champions League qualification.
"We'll fight to the end, (Champions League qualification) will be tough but we'll see what we can do," he added.