Brown's 80-yard touchdown run late in the game iced the initial victory of the 2011 season for the Colts, who improved to 1-13 with the triumph.
The victory was the first for the Colts without injured quarterback Peyton Manning since 1997.
Linebacker Pat Angerer contributed an interception and caused a fumble for Indianapolis.
Tennessee (7-7) was led by Chris Johnson, who ran for 55 yards and caught eight passes for 54 more.
Starting quarterback Matt Hasselbeck lasted three quarters while playing on an injured ankle, tossing for 223 yards with two interceptions before yielding to Jake Locker in the final frame.