The England striker reached 200 goals for the club on Tuesday night when he scored twice in the Champions League win over Bayer Leverkusen, with his double century coming in just 406 games.
He is fourth on the all-time record scorers list for the Red Devils and after overcoming a turbulent summer, where his future remained uncertain, Rooney looks a good bet to overhaul Jack Rowley, Denis Law and Bobby Charlton and become the club's most prolific marksman.
It is a quick turnaround in fortunes for the 27-year-old, who was the subject of two bids from Chelsea after reportedly asking former manager Sir Alex Ferguson for a transfer.
And Anderson, who made 69 appearances for United under Ferguson, believes he is looking back to his best.
"He's a United great and his record says that, 200 goals is great. He is only 40-odd short of Bobby Charlton and Denis Law, so he has to be talked about like that," the 57-year-old told Press Association Sport.
"He has had his ups and downs with the supporters but when it comes to matters on the football pitch, he gives 100 per cent and I am personally pleased he is back on board.
"Some supporters will remember the events of the summer but a lot of them will think of him for what he does on the football field and the great goals he has scored over the years.
"There will be a minority who perhaps won't look upon him fondly, but I think the majority will still love him.
"He had to knuckle down and that's what he did. He got back in the team, he is scoring goals and looking like his old self when he first arrived.
"He looks back to his best."
New boss Moyes was handed a difficult challenge, having to follow in the footsteps of Ferguson, but Anderson expects United to challenge for all four major trophies this season.
"Whoever took the job was going to find it tough because you've got big shoes to fill," he added, speaking at an event in Rotherham to promote Vauxhall's Youth Football Programme for which he is an ambassador.
"Sir Alex Ferguson won 20-odd trophies and David knew this, but the chance to become Manchester United manager is an opportunity not many people get and I can understand why he took it.
"He has to challenge on all fronts this season, that will have been the remit.
"Once you take that job, you expect to be challenging on for all four major competitions. Manchester United always have to be challenging."
- Sports & Recreation
- Sir Alex Ferguson
- Bobby Charlton
- Bayer Leverkusen
- Denis Law