Januzaj made a lasting impression in his first Premier League start for United two weeks ago, scoring twice in a comeback victory over Sunderland at the Stadium of Light.
Moyes believes that Januzaj has a support network that will keep him grounded, declaring that has more stability as a teenager than Wayne Rooney did during his formative years at Everton.
He told reporters: "He is probably the most level-headed young boy I have ever come across.
"I have had Wayne Rooney and Ross Barkley and all the other ones who have come through at different clubs. He is the most steady and stable. The boy is in really good order.
"He is certainly kept in check by his father and his family but mentally he is really good. At the moment I don't see any problems. Maybe in years to come you will come back and say, 'Do you remember this interview?'. But hopefully he goes on to fulfil everything we think he will do."
The 18-year-old is out of contract next summer, but hinted that his long-term future lies at Old Trafford in an interview earlier today.