Fellaini struggled to live up to his £27.5 million price tag in his first season at United and the Belgium international was even thought to be close to ending his Manchester nightmare following Louis van Gaal's arrival at the start of the 2014-15 Premier League campaign.
But the ex-Everton midfielder has since elevated himself into Van Gaal's first-team plans after returning from injury, with two impressive performances against West Brom and Chelsea in recent weeks.
Fellaini was on target as United drew 2-2 at West Brom, while the towering 26-year-old was effective as he nullified Cesc Fabregas in last week's 1-1 stalemate with the leaders.
And Moyes, who brought Fellaini to Manchester before he was sacked as manager in April, believes the Belgian can become a key player for United.
"It's true that players can take time to settle at a new club. I remember people telling me it took Patrice Evra and Nemanja Vidic a while. Players who became great players for United," Moyes wrote in his column for the Daily Mail.
"Marouane should have been given time too, because he's now proving what I always believed: that he can be an important player for United.
"I've watched him in the last two games and he has been first class. He's had an influence on the games, which is what you look to your bigger players to do. And I think he's played an important part in helping United gain some momentum going into the Manchester derby.
"There were times last season when things were tough, when we spoke. But I told him then that he would become a top United player if he continued to work hard."
Moyes added: "He can perform several jobs in a team but I think it helps him to have someone like Daley Blind sitting there in that more defensive midfield role.
"At Standard Liege he had [Steven] Defour or [Axel] Witsel sitting deeper and allowing him to get further forward, and at Everton he had players like Phil Neville.
"At United last season we lacked that kind of midfielder, and while Marouane has the physical ability to cover the ground he's definitely more effective in a game if he's accompanied by a defensive midfield player.
"So I'm delighted he's settled down now and he's beginning to be appreciated for what he is; a damn fine footballer who was definitely worth the effort as well as the money."
