The Belgium international, who moved to Old Trafford from Everton on deadline day, replaced Anderson in the 62nd minute to help United see out a comfortable 2-0 victory.
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"It was a fantastic debut and we won, which was most important," he told MUTV. "I was both nervous and excited. I thought I had to play my game and with my quality.
"I would like to say thank you to the supporters, it was good for me. Every time I will give my maximum for the club and for my team-mates.
"It was a good win, not hard or easy, just a good win. Manchester United want to win every game and we won today so it was good for our confidence ahead of the next game - because the next game is a big game, it is a Champions League game."
David Moyes was pleased Fellaini could make his bow and protect Robin van Persie's opening goal, before Wayne Rooney sealed the three points in injury time.
"I thought Fellaini played well today, hopefully all the supporters saw what he can do," the United boss added.
"We were delighted he got on, I wanted him to win the ball for us, give it to our good players so we could go and score goals.
"He’s more than that, but I thought today he came on and calmed it down when we were only winning 1-0. I had to make sure, even though they had ten men, that we didn’t do anything stupid and we didn’t."
Next up for the Premier League champions is a visit from Bayer Leverkusen on Tuesday night for their Champions League Group A opener.