After impressing on the Red Devils pre-season tour of the Far East and Australia, and then being involved in the Community Shield win over Wigan, Januzaj had to wait before being unleashed onto the first-team scene.
He finally made his Premier League debut against Crystal Palace on September 14, scored twice on his first start at Sunderland to turn the game on its head and has been a regular presence since.
United's supporters have been so enchanted by the youngster they have already come up with a pretty impressive song in his honour.
Not bad for an 18-year-old, who is still coveted as a potential member of the Belgium World Cup squad even though Januzaj has shown little inclination to commit his international future to any of the numerous nations who have a claim on his services.
"Adnan is doing brilliantly well," said Moyes.
"We took him on tour and since then he just kicked onto another gear.
"He was going past people on the inside and outside. All his decision-making was correct and he can score with both feet.
"He is still learning and we still have to be careful with him but he has gone on so quickly."
Januzaj's progress pleases Moyes because it fits in with the Manchester United tradition he is so keen to be a part of.
The Scot famously gave Wayne Rooney his debut, and also offered Ross Barkley his first taste of senior action at Everton.
And he knows how important youth development is at his current club.
"I see it written on the walls around the academy," said Moyes.
"I do want to be a manager who gives young boys an opportunity.
"I was like that at Everton - and Preston as well. I hope I am able to do that at Manchester United."