The two scoring strikes rallied the Colts from a 17-6 halftime deficit and sent them to a victory that put them in control in the AFC South.
Indianapolis climbed to 7-3 and moved three games in front of second-place Tennessee in the division race.
Luck drove the Colts 74 yards in 11 plays during the opening possession of the second half and Brown ended the march with a 6-yard scoring run that brought Indianapolis within four points.
Devon Wylie then fumbled the ensuing kickoff, which the Colts recovered at the Tennessee 20. Luck scored two plays later on an 11-yard run.
Another touchdown run of 11 yards by Brown with 3:01 to go put the Colts in front by 10 and allowed them to survive a touchdown pass by Ryan Fitzpatrick with 1:54 remaining.
Indianapolis recovered the following onside kick and ran out the clock.
Fitzpatrick, who inherited the starting quarterback role when Jake Locker was lost for the season with a foot injury, completed 22-of-28 passes for 222 yards. Tennessee built its early lead thanks to first-period touchdown runs of 30 and 7 yards by Chris Johnson.
Luck went 23-of-36 for 232 yards without a touchdown or interception.