Brown rushed 25 times for an average of 12.4 yards per carry and became the third player in school history to surpass 300 rushing yard and the first player since Rashaan Salaam had 317 yards against Texas on his way to winning the 1994 Heisman Trophy.
It was also Brown's third consecutive game of more than 100 rushing yards. Salaam was the last player to have three in a row, when he had nine straight in 1994.
Overall, Colorado (4-2, 2-0 Big 12 North) totaled 427 yards on the ground as Bobby Purify added 76 yards on just 10 carries.
Brown's 78-yard run gave the Buffaloes a 14-0 lead with 10:05 left in the first quarter. It came just 74 seconds after Robert Hodge found D.J. Hackett for an eight-yard score.
However, the Jayhawks scored 15 consecutive points on a safety, a 26-yard pass from Bill Whittemore to Marcellus Jones and a five-yard run by Whittemore four seconds into the second quarter to take a 15-14 lead.
Colorado then scored 21 straight points. Brown raced in for a 51-yard touchdown with 14:11 left in the second quarter and Derrick McCoy caught a 15-yard pass from Hodge to give the Buffaloes a 29-15 lead with 6:04 remaining.
Kansas (2-5, 0-3) moved the ball on its next drive, getting to the Colorado five. But safety Donald Strickland picked off Whittemore and raced 95 yards along the left side for a touchdown to make it 36-15.
The Jayhawks got within 39-29 with 6:30 left in the third on Whittemore's nine-yard run. But McCoy caught an eight-yard pass and Brandon Drumm raced in from 21 yards out to put the game away.
Colorado registered its highest point total ever against Kansas and the 82 combined points were the most in the 62-game history of the series.