Indy (8-8) went 75 yards in 84 seconds with no timeouts to avoid its first losing season since 2011, sending retiring linebacker Robert Mathis out with a win in his final NFL game.
Jacksonville (3-13) matched the second-worst record in franchise history after blocking a punt with 1:54 to go and breaking a 17-17 tie with 1:33 left.
Instead, Luck took the Colts right downfield for the score. He finished 24 of 40 with 321 yards, two touchdowns and one interception in a game full of milestones.
Mathis extended his league record of strip-sacks to 41 in the fourth quarter, two days after announcing he would retire.
Frank Gore ran 16 times for 62 yards, becoming the fourth player in league history to top 1,000 yards at age 33 or older. He's also the oldest to achieve the milestone since John Riggins in 1984 at age 35, and he's the first Colts to run for 1,000 since Joseph Addai in 2007 - ending the second-longest active streak in the NFL.
Luck finished with than 4,000 yards passing for the third time in four years on a day Hall of Fame executive Bill Polian was inducted into the Colts' Ring of Honor at halftime.
That combination put this game, with no playoff implications, in a secondary role.
But the Colts finally got themselves righted late in the first half, cutting the halftime deficit to 17-3.
THE WAITING GAME
T.Y. Hilton came into Sunday with the most yards receiving in the NFL and a chance to become the first Colts to win the receiving title since Reggie Wayne in 2007.
After padding his lead with six catches for 95 yards, he still wasn't sure if it would be enough to hold off his nearest competition, New York Giants receiver Odell Beckham Jr. and Atlanta receiver Julio Jones. Both played in late afternoon.
Both teams must now sort out their coaching situation.
Jacksonville has already fired Gus Bradley. After installing Doug Marrone as interim coach, the Jags broke a nine-game losing streak - the longest in franchise history - before this loss.
Indy's second straight playoff absence has put the futures of coach Chuck Pagano and general manager Ryan Grigson in jeopardy despite going (48-31) and never having a losing season during their five-year tenures.
Soon, all three will know whether they've done enough to retain their jobs.