Pryor, coming off a sophomore season in which he led the Buckeyes to an 11-2 finish, including a Rose Bowl victory over Oregon, completed 17-of-25 passes.
Brandon Saine added 103 yards on the ground and two scores for Ohio State, which has won at least a share of the Big Ten title in each of the last five seasons.
Brian Rolle ran back an interception for a score for the Buckeyes (1-0), who will meet nationally-ranked Miami-Florida next Saturday.
Ahmed Shakoor scored on a return of a blocked field goal for Marshall (0-1), which played its first game under new head coach Doc Holliday. The Thundering Herd finished 7-6 last year, including a victory over Ohio in the Little Caesars Pizza Bowl.
Brian Anderson completed 17-of-28 passes for 135 yards for Marshall.