Brown carried 35 times for 211 yards, one of which came on the decisive touchdown. The nation's leading rusher also scored from 10 yards in the second quarter and 15 yards in the fourth.
Missouri's overtime possession ended when freshman safety J.J. Billingsley forced Darius Outlaw to fumble. Safety Kory Mossoni recovered at Colorado's 12, sealing his team's 16th win in the last 18 meetings with Missouri.
The Buffaloes won for the fifth time in six games following a 1-2 start. They can clinch the Big 12 North title and earn a place in the conference championship game with wins over Iowa State and Nebraska.
Colorado (7-4, 5-1 Big 12 North) opened a 27-7 cushion on the first possession of the second half before Missouri scored three straight touchdowns to take the lead with 13:16 to play.
Freshman phenom Brad Smith threw a pair of touchdown passes to Justin Gage, but the Tigers (4-5, 1-5 North) lost for the fifth time in six games and fell to 3-50 against ranked teams since the 1989 season.