The Belgium international signed for the Premier League champions from Everton on deadline day for £27m, but has admitted that he feared a move may never materialise.
"Sometimes I thought there was no chance [of the move happening] but in the end I was so happy to join United," Fellaini said in a press conference.
"I came here to win trophies. For me it's a dream to join the biggest club in the world. I need to work hard for my place. There are a lot of good players here. I need to show my ability. I'll keep focused."
The 25-year-old could feature in United's Premier League clash against Crystal Palace tomorrow.