Steven Gerrard scored twice from the spot as Liverpool beat Man United 3-0 at Old Trafford on Sunday - and Moyes revealed after the game that he is finding his job difficult.
"I think the job was always going to be hard. Harder? Yeah, I would say so." he told reporters.
United saw Nemanja Vidic sent off for the foul that led to Liverpool's third penalty, which Gerrard missed, and 50-year-old Moyes questioned some of referee Mark Clattenburg's decisions.
"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 penalty 20 seconds into the second half made it an upward climb.
"They 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.
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Despite suffering another setback in a disappointing first season in charge at Old Trafford, Moyes has refused to concede defeat in the chase for 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.