Franklin's five-yard TD run up the middle with 4:35 left in the first quarter put Missouri (1-0) ahead for good, and he capped the scoring with a 10-yard scoring strike to Marcus Lucas in the fourth.
Franklin finished 17-of-26 for 129 yards with one interception while racking up 72 yards on the ground.
T.J. Moe hauled down six passes for 56 yards for the Tigers.
Miami-Ohio's Zac Dysert rolled up 194 aerial yards on 26-of-39 passing with no touchdowns and one pick-off.
Erik Finklea got the only TD for the Redhawks (0-1) on a 10-yard, third-quarter run.