The Belgium international completed a €32.5 million move from Everton on transfer deadline day and the Red Devils boss, who coached the midfielder for five years during their time together at Goodison Park, is extremely excited by his old charge's arrival.
"Marouane is not someone you want to be playing against," Moyes told reporters. "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."
United had initially made a combined bid for Fellaini and Toffees team-mate Leighton Baines earlier in the summer that was both rejected by Everton and described as "derisory".
The English champions ultimately failed in their bid to land Baines but managed to sign Fellaini just before the close of the transfer window.
However, the Old Trafford outfit also picked up 17-year-old Saidy Janko from FC Zurich, but only confirmed the transfer on Tuesday.
The Swiss defender is the club's third signing of the window following Fellaini's big-money arrival and that of full-back Guillermo Varela from Penarol.