The 25-year-old Belgian midfielder's move from Everton to be reunited with former Goodison Park boss David Moyes was completed late on Monday night.
United's new manager had been linked with several players including Toffees left-back Leighton Baines, Athletic Bilbao's Ander Herrera and Real Madrid's Fabio Coentrao, but his summer-long efforts ended with Fellaini being his only high-profile recruit.
The scarcity of signings provided much of the post-match debate following United's 1-0 defeat at Liverpool last Sunday but Fellaini, who had Old Trafford on his mind before United made their decisive move, is confident of achieving success with his new club.
After helping Belgium to a 2-0 World Cup qualifying victory over Scotland at Hampden on Friday night, Fellaini turned his attention back to United.
"It is a big step for me," he said. "But I have played five years in England, I know the league and I know the players so I don't think I will have a problem with this move.
"Of course I look forward to it, I am playing with big players so I need to work hard for my place.
"To win a trophy with Manchester United and play a lot of games, that is important.
"I watched the game (against Liverpool) and in the second half the team played very well and had a lot of chances but couldn't score, but football is like this.
"It will be tough, every year is tough for the big teams but I think Manchester United has the quality to win the championship so I look forward to this."