Fellaini completed his £27.5million move from Everton to United just before the transfer window closed last night.
"I have worked with Marouane for five years and I'm delighted he has decided to join Manchester United," said Moyes.
"He is a player with great ability and strength and I think he will make a real difference to our squad."
Fellaini could have joined United earlier this summer for just £23.5million as he had a release clause in his Everton contract, but that expired on 31 July.
Fellaini said: "I am thrilled to sign for Manchester United.
"I have known the manager for many years now and I have a lot of respect for his way of working.
"When I was given the opportunity to work with him again, I jumped at the chance.
"I had a fantastic time at Everton and I'd like to thank all the fans and staff there for everything they have done for me.
"But being a Manchester United player is something that every player dreams of and I believe that I can make a real contribution to the team's efforts to win trophies in the future."