Missouri (6-0, 2-0 Southeastern Conference) led by 18 points in the first half before Georgia (4-2, 3-1) cut the lead to 28-26 in the fourth quarter.
The Tigers answered the challenge with two late touchdowns despite losing Franklin to a shoulder injury.
After Franklin was taken to the locker room, freshman backup Maty Mauk threw a lateral to Sasser, who stopped and tossed a deep to the end zone for Washington. Washington, who outfought cornerback Damian Swann for the catch, had 115 yards and two touchdowns.
James Ponder's interception of Aaron Murray's pass with 4:25 remaining set up Henry Josey's 7-yard touchdown run to end Georgia's hopes.