The attacking midfielder is keen to stay at Old Trafford for a long time and is confident the club will be successful if it shows patience with its new manager
Since making his first-team debut in August's Community Shield victory, the youngster has become a regular under the former Everton boss this season, scoring twice in 10 Premier League games.
However, Moyes has struggled after succeeding Sir Alex Ferguson in the summer, with his side sitting 10 points behind leader Arsenal in eighth place, but Januzaj says that fans must be patient.
"He will win trophies," the 18-year-old told MUTV.
"Don't forget, it is a new coach. Everything is new. You have to give it time.
"I think he is a good coach. All the players are ready to work hard as well and everything will come."
Januzaj, who arrived at United in 2011 after spending time in the FC Brussels and Anderlecht academies in his native Belgium, says that wants to stay at Old Trafford for "a long time".
"I am happy here," he enthused. "When I moved to Manchester I wanted to play with the first team. Now I want to stay here, play for a long time, if I can, and be the best player I can be."