The 18-year-old struck twice in six second-half minutes to seal United's win and cap his first Premier League start with a 2-1 victory.
He followed in the footsteps of Ruud van Nistelrooy scoring twice on his league debut and Moyes hailed his influence.
"I remember giving Wayne [Rooney] his debut, and Ross Barkley, and Adnan is certainly that quality. He is a top, top player," Moyes told Sky Sports.
"I would've played him two or three weeks ago, but for different reasons it didn't quite work. I thought today was the right time, he's been so good it was hard to stop him.
"He's going to be a great talent, he's a special talent, he got goals, he's got ability on the ball, but we'll keep his feet on the ground and make sure he gets on with it.
"We won't let him get ahead of himself and we'll keep trying to do the right things with him."
United made the worst possible start to the game as Craig Gardner capitalised on some sloppy defending to give the hosts the lead but Moyes insisted he was happy with the overall performance from his side.
"I thought it was just a good performance, even a 1-0 down I thought we played really well," he added. "We made a real hash of the first goal we gave away but apart from that I thought we played some really good stuff.
"I'll not be getting too excited because I've won, just as when I was losing I wasn't getting too low either, a lot of people were but I wasn't.
"I know the quality in the players I've got in the dressing room."
United's win moves them back into the top half of the Premier League table with 10 points from their opening seven matches.