Franklin rushed for 142 yards and tossed for 132 more as the Tigers (8-5) jumped out to a 31-7 lead in the first half and cruised to the victory.
Missouri finished the season with five consecutive wins, including four straight in the Big 12 Conference. The school will switch to the Southeastern Conference next season.
The Tar Heels (7-6) fell despite 317 yards and three touchdowns from Bryn Renner. The North Carolina signal-caller was intercepted once.
Jheranie Boyd made three grabs for 95 yards and a score.