Moyes signed the Belgium international from Everton last summer after the duo spent five years together at Goodison Park.
But Fellaini has struggled to make an impact at United and he is yet to score for his new club while his passing has at times been woeful.
Spanish newspaper AS gave him 1 out of 10 for his performance against Bayern Munich in the UEFA Champions League on Tuesday.
But Moyes says he believes Fellaini is beginning to find his feet after a difficult first season at Old Trafford.
"It's not an easy move coming to Manchester United, as I can tell you," Moyes said.
"For Marouane, I think it's proved equally as difficult, but I have to say I think he has done a lot of good things and he has improved.
"I actually thought Marouane got better as the game went on (against Bayern). We needed him and I thought he played very well.
"He did well in the Villa game too. I think he is getting much closer to scoring.
"We bought Marouane Fellaini not to play every single game in the Premier League. We have bought him for several games. He did a great job at West Ham, against Aston Villa and against Bayern."