The former Gunners boss is backing Sir Alex Ferguson's replacement this summer to do a "brilliant job" at Old Trafford
Graham, who played for United from 1972-74, believes that the two coaches have very similar qualities which will make the transition at Old Trafford easier for the club.
Moyes failed to win a single piece of silverware during his 11-year tenure at Goodison Park, prompting some to question his ability to step into his compatriot's shoes, but the ex-Gunners boss believes he will do well.
"It is the natural progression in his career," Graham told The Daily Mirror. "He has served his time as an apprentice at Preston and then at Everton - where he did a brilliant job.
"I believe he is the perfect choice to take over. United could have turned to a foreign manager, but I am glad they stayed British.
"There are a lot of similarities between Moyes and Sir Alex. Moyes has the same hunger and ambition - and he has that drive that is so common from people from the west of Scotland."
Moyes has been in management for 15 years, starting his career at Preston in 1998, before taking over from Everton in 2002 and Graham believes his track record of longevity at clubs proves he will succeed in his new role.
"He has drive to be successful," continued Graham.
"He is also a realist and knows there will be inevitable comparisons between him and Sir Alex, especially if United lose a few games. But he will ready for that. He will be prepared - and it will not concern him about handling big names.
"There were star players at Everton when he first arrived. It won’t be a problem for him. He will relish the challenge and it would be no surprise to me if he won the title next season."