Danny Welbeck's first-half brace and Tom Cleverley's first Premier League goal in nearly a year were enough for United to stave off the threat of a hat-trick of defeats.
Darren Fletcher's return after almost a year out with a chronic bowel complaint put further shine on an impressive day for the visitors.
But Moyes was not happy at the treatment Januzaj received from the home players, and fears the 18-year-old is going to get injured if something is not done to prevent the rough-house tactics.
"Just a bit," said Moyes, when asked if he felt Januzaj was being kicked.
"In every game, I don't know how many tackles there have been on him.
"That is why I am annoyed with fourth officials and referees. He is definitely not getting protected and before you know it the boy will get a sore one.
"It is terrible what they (the officials) are allowing to go on at the moment. They (opposition players) are queuing up to see who can kick him next.
"(Matthew) Lowton was booked but there were that many tackles on him from different players.
"It is a recurring theme and it's the referees' job to sort it out. You would hope they would recognise that.
"Adnan will just get up, walk away and get on with it. That is one of his biggest attributes."
Not surprisingly, Villa boss Paul Lambert rejected the assertion of his team being over-physical.
"My team are not a physical team," said the Scot.
"We are a young side and maybe sometimes over exuberance gets the better of them but I have never sent anyone out to kick them."
It was Januzaj's bullet header that provided the rebound for Welbeck to tap United in front.
The England striker, selected in the absence of Robin van Persie, was netting for the first time in the Premier League since the opening day of the season and quickly added another when he swept home Antonio Valencia's cross.
"It is really important for Danny," said Moyes.
"He is a forward and he needs to get himself goals. He got two poacher types today and he could have had another.
"With Robin out he stepped up to the mark. When you get opportunities, you show what you can do. Danny certainly did that."
There was never any doubt over the outcome once Cleverley had struck from Wayne Rooney's lay-off, allowing Moyes to end Fletcher's near 12-month absence with a chronic bowel complaint, that has plagued him for almost three years.
"If anybody returns from an illness to play professional football on this kind of stage for a club like Manchester United, it shows a lot about the boy himself," said Moyes.
"He has had to be on his own, he has been in and out of hospital, he has had operations.
"Today was a big message to a lot of people - keep believing.
"If you are not well then keep believing that you can get yourself right."
Fletcher declared himself satisfied at his comeback.
"I always believed I would come back," he told MUTV.
"Other people around me were trying to make me think otherwise, but I stayed strong and believed I would get back.
"Playing a game like that, and getting an ovation like that, makes it all worthwhile."
Moyes confirmed Shinji Kagawa was still struggling with the after-effects of a sickness bug, whilst Javier Hernandez was ruled out after taking a kick on the head in training.
Although the result has not done much to alter United's league position, it at least provides a platform for them to kick-on, starting in Wednesday's Capital One Cup quarter-final at Stoke.
"We are getting closer but it has to be more consistent," said Moyes.
"We could have scored more goals but the all-round performance was much more like us and I am pleased with that."
Lambert was less pleased, with striker Christian Benteke a particular disappointment as his run without a goal reached 10 games.
"He will come through it," said Lambert.
"His endeavour was there. He is not getting the breaks he had last year but he will go through stages like this in his career."
Source: PA
Source: PA