Fellaini followed Moyes from Everton to Old Trafford last summer but it has been a difficult first season at the club for both men, with United struggling for consistency and set to miss out on UEFA Champions League qualification.
However, Fellaini reckons it was always going to be tough for United to adjust to life without Sir Alex Ferguson and he has thrown his support behind Moyes.
The Belgian midfielder has no doubts that Moyes has the qualities to get United back on top and hopes the fans will remain patient.
"What makes United different is that they stand by managers," Fellaini said. "You look at the career of Sir Alex Ferguson and he managed the club 26 years.
"You have to give David Moyes time as well. He is a strong character. That's why he can take the pressure. Given time he will be good for Manchester United.
"Man United had the same manager for 26 years - to change that is not easy for anyone. The supporters and everybody else has to give him time."
Fellaini has worked under Moyes since being brought to Everton from Standard Liege in 2008, and he is grateful for everything the manager has done for him.
"I can only say a huge thank you because I learned so much from him," Fellaini said.
"When I arrived at Everton he spoke to me about English football, the life and the style of the country. Also at Man United he spoke to me about life there as well. It is important because we have football but we also need a life as well.
"I played a lot of games for Everton and had some very good moments there. I could never say something bad about him because he has done everything for me. When I played bad he told me. When I played well he told me. When I have a bad demeanour on the pitch he tells me.
"When it's bad off the pitch he tells me that as well. It's important for me that when I am good he tells me. And when I am bad he does the same.
"Last season when I was suspended for Stoke City [after a headbutt] he was very hard on me. He made me go back out and train alone with him. I was like his son and I had been bad so I took my punishment."
Fellaini added: "He is the most important person in my career. I could never say something bad about him because he has done everything for me.
"He has done so much or me that of course I want to repay him at Manchester United."