It has been a tough baptism for Moyes as Sir Alex Ferguson's successor, with a string of disappointing results sending the Red Devils spinning down the Premier League.
Sunday's 3-0 win at Aston Villa did suggest a brighter future, although Moyes is mindful his side have gained similarly encouraging results before, only to immediately lose their momentum.
However, Januzaj, one of the players upon whom the 'new' United must be built, feels Moyes will continue with the club's trophy-winning traditions.
"He will win trophies," the winger told MUTV.
"Don't forget, it is a new coach. Everything is new.
"You have to gave it time.
"I think he is a good coach. All the players are ready to work hard as well and everything will come."
Januzaj has been a more regular member of the United squad in recent games, having marked his first appearance with two goals at Sunderland in October which turned the game at the Stadium of Light on its head.
He has already been rewarded with a five-year contract extension, allaying fears United could lose another of their promising stars, as happened with French midfielder Paul Pogba, who is now doing such good things for Juventus.
Yet moving was never on the agenda for Januzaj, who created United's opening goal at Villa Park when he crashed a header onto the post.
"I am happy with here," he said.
"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."
For Januzaj that means emulating another winger who left his home for England at a very young age - Cristiano Ronaldo - and became one of the world's best players.
The example of Ronaldo proves what can be achieved.
But there are others, much closer to Manchester. And Januzaj feels he is learning from the best in Ryan Giggs, whom, he is convinced, still has much to offer despite hitting his 40th birthday.
"Ryan is a legend, totally unbelievable," said Januzaj.
"If I don't know what to do on the pitch, I just go to Ryan and ask him what I need to improve on.
"He helps me a lot - and he can still carry on for a bit yet.
"He is 40 but I think he could play for another five years."
Moyes has little time to reflect on United's success at Villa Park.
After a quick trip to Leeds, where he was part of the United delegation that attended the presentation of the lifetime achievement award handed to Sir Alex Ferguson at the BBC Sports Personality of the Year awards, he was back over the Pennines to plot the downfall of Stoke in Wednesday's Capital One Cup quarter-final.
United will need to take a check on Shinji Kagawa, who has still not fully recovered from having to have his stomach pumped a fortnight ago, and Javier Hernandez, who missed yesterday's game after taking a kick on the head in training on Saturday.