Cleverley, 21, impressed many pundits with his second-half display during United's Community Shield victory over Manchester City at Wembley on Sunday afternoon.
United manager Sir Alex Ferguson has been searching for a central midfielder since Scholes retired at the end of last season, but Evra insists that Cleverley deserves a chance to fill the role.
"Tom is a fantastic player, and I hope this season that he will get a lot of opportunities," said the French full-back.
"In pre-season in America he had some amazing games, particularly against Barcelona.
"He's a quiet lad, he's a bit like Scholesy in that way. He never talks to much, he just does his job on the pitch. He's a very private guy.
"Bringing young players through is what we do here. Players like Tom are the future."
Cleverley has been called into the England squad for Wednesday's friendly with Holland.