Brown also scored on a 10-yard run in the first quarter and a 15-yarder in the fourth, giving him 17 rushing TDs this season. He had 35 carries on his seventh consecutive 100-yard day, and Bobby Purify also hit the century mark for the first time this season with 119 yards on 19 carries.
Colorado (7-3, 5-1 Big 12) amassed 575 yards in total offense to hold onto its North Division lead entering next week's home game against Iowa State, which entered the day one-half game back.
"In November, they're all hard,'' Colorado coach Gary Barnett said. "Right now, we're just biting and scratching and finding a way to win.''
But first, the Buffaloes got a huge scare from Missouri (4-6, 1-5), which rallied from a 27-7 third-quarter deficit with three straight touchdowns, then tied it to force overtime.
"We played an amazing second half,'' running back Zack Abron said. "If we had come out with that same energy in the first half, it probably wouldn't have been a close game.''
After escaping, Colorado players and coaches talked about their resilience.
"We knew that it was going to happen during the season,'' quarterback Robert Hodge said. "It was good to get that under our belts, a win where we came from behind.
"It's a fight and we know that coming in, and to pull off a win under those circumstances is really big.''
Brown, who has 1,617 yards this season, carried six times for 24 yards in the extra period, including a 12-yarder to the 4 yard-line.
On Missouri's overtime possession, Darius Outlaw caught a pass on third-and-10, but he fumbled after being hit by J.J. Billingsley. Kory Mossoni recovered for the Buffs.
"I hate that it happened and I wish that it had never happened,'' Outlaw said. "But I fumbled and lost the game. Game over.''
Mossoni said he was about to hit Justin Gage when he saw the ball.
"I was on my hands and knees scrambling to get that ball,'' Mossoni said. "That's the fastest I've ever crawled.''
It was the second straight heartbreaking loss -- by the same score -- for Missouri, which has lost 18 straight against ranked teams. The Tigers trailed 27-14 at Iowa State last week and took the lead 28-27 before losing 42-35.
"In this darkest moment, they're probably saying in their minds 'What do we have to do to win?''' coach Gary Pinkel said. "Well, guess what, we've got to do a little more. It's not real complicated.''
Abron ran for two touchdowns, including a 1-yard run that forced overtime, and Gage caught two touchdown passes from Brad Smith for Missouri.
Colorado scored on its first five possessions to take a 27-7 third-quarter lead.
After Missouri rallied to take a one-point lead, Colorado regained the lead on a 15-yard run by Brown and a 2-point conversion pass from Hodge to Quinn Sypniewski with 4:43 to go. Abron scored his second touchdown of the game with 1:55 left, and Mike Matheny added the extra point to make it 35-all, capping an 11-play, 67-yard Missouri drive in only 2:38.
The biggest play of Missouri's comeback was a 43-yard interception return for touchdown by Taurus Ferguson late in the third quarter, cutting the gap to 27-21. Missouri took its first lead at 28-27 on a 10-yard pass from Smith to Gage with 13:16 remaining.
"We threw one to them and all of a sudden we're in a dogfight that we didn't think we'd be in,'' Barnett said. "They really didn't do anything different, they just completed some balls and we didn't make a couple of plays.''