Brown ran for 79 yards on 10 carries with a 51-yard score and made two catches, including a 33-yard touchdown pass from Andrew Luck for the fourth-seeded Colts (10-5).
Luck completed 26-of-37 passes for 241 yards and Adam Vinatieri kicked three field goals.
The fifth-seeded Chiefs (11-4) had their two-game winning streak snapped as Alex Smith was held to 153 passing yards on 16-of-29 efficiency.
Jamaal Charles ran for 106 yards with a score on 13 carries.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Millions of Getty images now available for free via embed tool