The 18-year-old Belgian played the final 23 minutes against Crystal Palace at Old Trafford and did not look out of place.
He even played a part in United's second goal nine minutes from time when he was chopped down by a defender as he sprinted towards the penalty area, and Wayne Rooney subsequently curled home the resulting free-kick.
Moyes told United's official website: "Adnan's ahead of schedule.
"I would have introduced him earlier if the games had been suited, but the start of the Premier League has been a bit more difficult.
"I thought, if I got the opportunity against Palace, I'd introduce him and I considered playing him from the start.
"His performance when he came on shows that would have been merited. He did a lot of good things, he moved with the ball well. The boy's going to be a good player."
And talking about his league debut, Januzaj told MUTV: "I felt great and shocked because I played for about 25 minutes and that was my first game.
"I think I did some good things but it is only my debut so I have to keep doing well on the pitch."