The 18-year-old Belgian was handed his first Premier League start at the Stadium of Light and repaid his manager's faith with two wonderful goals as United hit back from Craig Gardner's early strike to win 2-1.
Moyes is convinced that Januzaj has the potential to become a great player and insists there is no chance of the youngster getting carried away by his first taste of success.
"I remember giving Wayne his debut, and Ross Barkley, and Adnan is certainly that quality. He is a top, top player," Moyes told Sky Sports.
"My plan was to play him three or four weeks ago but I never quite got the opportunity and for different reasons I didn't pick him, and I thought today was the right time.
"He has been so good, it was hard to stop him.
"He is going to be a special talent. He has got goals, he has got ability on the ball, but we will keep his feet on the ground here."
Januzaj will be out of contract at the end of the season and his breakthrough performance could lead to increased attention from other clubs, but Moyes does not see why he would be interested in leaving.
"I am not too worried about that. I think every young boy wants to play at Manchester United, I have always thought that."
It was an important win for United after last weekend's defeat by West Bromwich Albion and Moyes was pleased with the performance of his side, although their defending was poor for Gardner's goal and David de Gea was required to produce a magnificent save from Emanuele Giaccherini's header when Sunderland were leading 1-0.
Moyes said: "It was an incredible save. I said to him at half-time that had kept us in the game.
"We made a real hash of the first goal we gave away but apart from that I thought we played some really good stuff today."