The Belgian international completed a £27.5 million move from Everton on transfer deadline day, and Moyes, who managed Fellaini for five years at Goodison Park, has hailed his arrival.
"Marouane is not someone you want to be playing against," he said.
"He has attributes other people don’t have. And with those qualities he has, I am sure he will be a big player for Manchester United."
Everton made public a £28m combined bid for Fellaini and left-back Leighton Baines earlier in the window, which was rejected and described as "derisory".
United went on to pay £500,000 more than that figure for their new midfielder alone, with the pursuit of Baines ending in disappoinment.
The Old Trafford outfit also announced the signing of 17-year-old defender Saidy Janko from FC Zurich on Tuesday.
The Swiss teenager is the club's third signing of the window following Fellaini's big-money arrival and that of young full-back Guillermo Varela from Penarol.
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