Javon Pickett, Jarron Coleman and Ronnie DeGray III added 12 points each as the Tigers (11-20, 5-13 SEC) snapped their six-game losing streak.
Braelen Bridges scored 19 points, Kario Oquendo had 17 and Aaron Cook added 17 for the Bulldogs (6-25, 1-17), who finished their regular season with an 11-game losing streak.
Missouri outrebounded Georgia 43-24 and earned extra possessions with 14 offensive boards.
Georgia built a huge advantage in free-throw attempts, 33-15, but the Bulldogs missed 12 shots from the line.
The Bulldogs raced to an early 16-2 lead, triggered by Cook's two steal-and-dunk sequences. Missouri turned the ball over four times in the four minutes and started 1-for-7 shooting from the floor.
Amari Davis finally settled the Tigers with a 3-point play. That started an 11-4 surge that cut their deficit to seven points.
But Georgia responded with an 8-2 run, with Bridges scoring six points, to extend its lead to 35-21.
Missouri had just eight players available for the game and early foul trouble created additional strain in the first half. Davis missed time after picking up his second personal foul with 9:29 left and Brown sat until the 17:31 mark in the second half after picking up his third with 6:44 left.
The Bulldogs went nearly 3 1/2 minutes without scoring late in the first half, but Oquendo's driving layup gave them a 39-30 halftime lead.
Coleman hit three 3-point jumpers and Brown hit one from beyond the arc as the Tigers pulled ahead of Georgia 54-52, with 10:48 left with a 24-11 rally.
With Pickett scoring four baskets and two free throws by attacking the basket, the Tigers pushed their lead to 64-54.
Georgia cut that lead to 69-64 with 3:17 left but the Bulldogs could get no closer.