Anderson joined Fiorentina on loan during the January transfer window after starting just two Premier League games for United this season.
The 25-year-old does not believe he was given enough opportunities to impress at Old Trafford over the past few months, but respects Moyes' decisions.
The Brazilian midfielder has made three appearances since linking up with Fiorentina and is hoping that he can earn a long-term deal with the Serie A club.
"To be honest, I've not had the chance," Anderson told The Sun regarding his time at Old Trafford under Moyes. "He gave me one chance or a half, but not a lot.
"He is the manager and I have to respect him. He treated me well and he has a lot of players.
"I have to play and have my chance. It is better for me at Fiorentina.
"The manager wants me and has shown his trust. I can show him my football.
"Even though the contract is only on loan, I want to stay for a long time.
"Fiorentina are a great team with an ambitious project. I want to win in Florence like important players of the past like Edmundo and Batistuta.
"Everyone knows Fiorentina can progress a lot in Italy and in European competition."
Fiorentina currently sit fourth in Serie A, lost 2-1 to Udinese in the first leg of the Coppa Italia semi-final on Tuesday night and face Esbjerg in the last 32 of the Europa League later this month.