The Scot brought the Belgian into the first team last season, a year in which the youngster’s performances were one of the brighter spots in a difficult period at Old Trafford.
Moyes, now in charge at Real Sociedad, believes that Januzaj has the talent to go a long way at United.
"Some youngsters get overawed. Not Adnan. I realised that when he came to train with the first team soon after I joined," Moyes told FourFourTwo.
"I decided to take him on tour with the first team within minutes of watching him. He wasn't timid but hungry to succeed. He was one of the tour's bright spots.
"He possesses a level of self-assured arrogance, which is a positive as he's not too arrogant. On the contrary, he's a bright lad who is mature for his age.
"He's a wonderfully gifted player with great balance and the ability to go past people. When I see his poise on the ball, he has characteristics of Johan Cruyff."