Mayfield, the first overall pick in the 2018 NFL Draft, led Oklahoma to an impressive 34-6 record. Comparatively, Jones' college record with Duke was just 17-19.
"Some people overthink it," Mayfield said. "That's where people go wrong. They forget you've gotta win."
He added, "Either you have a history of winning and being that guy for your team, or you don't."
The Browns went 6-7 with Mayfield as their starter last season, finishing with an overall record of 7-8-1 after going a combined 1-31 in the two years prior. Cleveland has finished over .500 just twice since 2000.
Jones, meanwhile, has impressed during his two preseason contests but will begin his first NFL campaign as Eli Manning's backup.