L'Damian Washington caught seven passes for 115 yards and two touchdowns for the Tigers (6-0, 2-0 SEC), who broke Georgia's (4-2, 3-1) 15-game home winning streak.
Franklin left early in the fourth quarter with a sprained right shoulder and did not return. Missouri said X-rays were negative.
The Tiger scored 21 points in the second quarter and another 13 in the fourth to keep the game out of the Bulldogs' reach.
Georgia took the field with a depleted offense. Four key receivers and running backs were on the sidelines with injuries suffered in last week's clash with Tennessee.
Aaron Murray completed 25-of-45 passes for 290 yards and three touchdowns. He was intercepted twice.