The ex-Everton boss officially starts work as United's new manager this week having replaced Sir Alex Ferguson.
Graham believes that the similarities between Moyes and his predecessor make him the right man for the job.
"It is the natural progression in his career," Graham told The Mirror. "He served his time at Preston [North End] and 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'm glad they stayed British. There are a lot of similarities between Moyes and Sir Alex.
"Moyes has the same hunger and ambition, he has that drive that is so common from people from the west of Scotland."