Brown and Ross both made three 3-pointers, sending Missouri to an eight-point halftime lead that steadily grew in the second period.
Alex Oriakhi chipped in 13 points and eight rebounds for the Tigers (13-3, 2-1 Southeastern Conference). Missouri played without star forward Laurence Bowers, who is suffering from a strained ligament in his right knee.
The Tigers, having won three out of four, will take on No. 9 Florida in a road game on Saturday.
Georgia (6-10, 0-3) was led by 23 points by Vincent Williams.