Moyes took charge at Old Trafford earlier in the summer and midfielder Fellaini followed in his manager's footsteps on deadline day by making the move from Everton.
Round, who also worked under Moyes at Goodison Park, is convinced that Fellaini has the quality to shine for United and says his versatility will also be a tremendous asset.
"I am sure Marouane will settle in very quickly and be an integral part of United's success in the future, I really do," Round told the Manchester Evening News.
"When David came here he looked at the squad and thought one of the players we needed to bring in was someone with those physical attributes in the middle of midfield who could give us strength and aggression in certain games and maybe allow one or two of the others to play.
"That, coupled with his ability to score, made him feel Marouane would be a really good acquisition for United.
"When you get to work with him every day, you get to realise that technically he is excellent. His chest control is arguably one of the best in the world. You can bang the ball at him from anywhere and he will bring it down on his chest and bring others into play.
"His feet and his technique work are really good. He is aggressive and tackles. He also has great fitness stats.
"When you are building a squad to compete on all fronts you don't want every single player to be the same. You want options and differences. In a certain game it might be Tom Cleverley and Carrick in midfield, it might be Fellaini and Carrick, or Fellaini and Anderson. There are different permutations.
"Marouane was ready for this step up. He'll fit in well with the professionalism here and relish the challenge.
"He is a 10-15 goals a season man and that is good from midfield.
"He can play number four, sat in front of the defence, he can play an eight in the centre doing box-to-box and can play a number 10 where he drops off the front man. That's valuable when the range of games is so varied."
The scrutiny on Moyes has increased since he replaced Sir Alex Ferguson as United manager but Round insists he can cope with the pressure and has the hunger to keep the club driving forward.
"Without a doubt David was ready for this job," he said. "I had five years at Everton with him and he impressed me every day. There is not a day goes by where you take your foot off the gas with him. He is 100 per cent every day with an incredible work ethic.
"He also has an incredible desire to succeed. He now has the stature and presence to move to a club like this. I really do believe he will be a big success here.
"Now I have been in this club and environment I can understand why Sir Alex and the board appointed him, to keep that continuity, work ethic, drive and feeling about the club going.
"It was very important to them and I can understand now that I have been here and lived it. They were made for each other."