Delano Howell scored on a 61-yard return of a blocked field goal for Indy (4-1).
Luck led his ninth career fourth-quarter comeback by going 16 of 29 for 229 yards, beating Russell Wilson in the first matchup between last season's NFL rookie of the year runner-ups.
Indy trailed 12-0 early, took the lead after Howell's return, then rallied again to take the lead for good on Brown's fourth-quarter TD run. The Colts sealed it with a 2-point conversion pass and a late field goal.
Wilson finished 15 of 31 for 210 yards with two TDs, one interception and ran 13 times for 102 yards.