Moyes was lured to Old Trafford in the summer to oversee a new era following Sir Alex Ferguson's retirement after 26 years in charge but the transition has been a testing one.
United lie 10 points behind leaders Arsenal in the Premier League but teenage winger Januzaj, who has become a key member of Moyes's new-look side, is certain the former Everton boss can honour the club's trophy-winning traditions.
"He will win trophies," Januzaj 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 marked his first appearance with two pivotal goals at Sunderland in October and has already been rewarded with a five-year contract extension.
"I am happy here," he added.
"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."