Moyes said United are "beginning to laugh at referees" for their "terrible" decisions.
He added that they are "playing referees as well as the opposition".
The first-leg defeat was United's third loss in a week - the first time that has happened since 1992.
The Premier League champions were beaten 2-1 at home by Tottenham at the start of 2014 and followed that up with elimination from the FA Cup at the hands of Swansea, which was also played at Old Trafford.
The former Everton manager was also irked by a decision during the Spurs defeat, stating it was "scandalous" United were not awarded a late penalty.
Moyes, who succeeded Sir Alex Ferguson at United at the end of last season, said he had the pedigree to guide the club through "difficult" times.
"I have done it before, I'm experienced and I just sense that it's just on the verge of turning," he added. "I sense that we are about to get a bit better.
"If you are going to be in this job, then that's what happens. You have got to win and in the last few games, I have not won.
"We are going through a difficult time. It's part of football management, it's never going to be an easy ride."