Moyes, who helped nurture the talents of Wayne Rooney and Ross Barkley at former club Everton, must guide Januzaj through the early stages of his career following a two-goal contribution on his first Premier League start at Sunderland a fortnight ago.
The 18-year-old's international credentials were swiftly thrust into the spotlight, given he could play for Belgium, Albania or Turkey, while a controversial residency rule means he could be eligible for England.
That scenario has divided opinion but Moyes insisted the sudden attention had not appeared to trouble Januzaj.
"He is probably the most level-headed young boy I have ever come across," said Moyes.
"I have had Wayne and Ross 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."