Moyes handed Januzaj his Manchester United debut, with the youngster repaying his manager's faith in him by scoring twice against Sunderland on his first start.
The Scot then took Januzaj on loan when he took control at Sunderland, though he was unable to save the 'Black Cats' from relegation to the Championship last season.
Speaking following Moyes' appointment as Slaven Bilic's successor, Januzaj, now of Real Sociedad, said: "He's a good manager. Obviously it's been difficult for him too. I'm happy for him because it's a nice move and I hope everything goes well for him.
"He likes to play good football. It's not easy sometimes, it depends on the squad. People have to trust him because he's done it at Everton for a long time."