Luck drove the Colts on a 13-play, 69-yard drive late in the game, capped by Adam Vinatieri's decisive 43-yard field goal with 5:58 remaining.
He completed 30-of-48 passes with a pair of touchdowns en route to the NFL mark for first-year signal-callers as Indianapolis (5-3) won its third straight.
Fellow first-round draft choice Ryan Tannahill, meanwhile, finished 22-of-38 for 290 yards with a touchdown for the Dolphins (4-4), whose three-game winning streak was snapped.
Exploding whale video goes viral on Internet
Sign language interpreter at Mandela service called out as fake on Twitter