Luck finished with 205 yards passing to break Cam Newton's year-old rookie record of 4,051 yards in a season.
The fabulous freshman also extended his own rookie record for fourth-quarter comebacks to seven by leading his team downfield in the closing minutes.
Darius Butler returned an interception 32 yards for their other touchdown, helping the Colts (10-5) join the Miami Dolphins of 2008 as the only teams in NFL history to win at least 10 games after losing 14 or more the previous season.
Most of those wins have come under interim coach Bruce Arians, who has filled in admirably while Chuck Pagano underwent treatment for leukemia. Pagano could rejoin the Colts on Monday and coach for the first time since September in their regular-season finale against Houston.
Jamaal Charles had 226 yards rushing for the Chiefs (2-13), including an 86-yard touchdown scamper on the first play of the second half. It was Charles' second 200-yard game of the season, and came one week after he managed 10 yards on nine carries against Oakland.
Charles joined the Titans' Chris Johnson and former Lions great Barry Sanders as the only players with three runs of 80 or more yards in a season. His third career 200-yard game also set a franchise record, which he had shared with Larry Johnson.
Peyton Hillis added 101 yards rushing for the Chiefs, but they didn't get a whole lot of help from Brady Quinn, who was just 19 of 22 for 162 yards and two interceptions.
The Colts also could have reached the playoffs if the Steelers lost or tied, but preferred to take care of it themselves, and before their regular-season finale against Houston.
It looked good for them right from the start.
On the Chiefs' fifth offensive play, Quinn threw woefully behind Dexter McCluster who wasn't even looking for the ball and Butler picked off the pass in stride, taking it untouched to the end zone for a 7-0 lead.
Kansas City managed to get a field goal out of Ryan Succop, but he missed his next try and the Colts went the other direction to set up the first of Adam Vinatieri's two field goals.
Charles fumbled in the red zone later in the second quarter the biggest blemish on his otherwise spectacular day. That gave Luck and the rest of the Colts offense enough time to move into Vinatieri's range, and his 36-yard field goal in the closing seconds made it 13-3 at halftime.
Charles electrified a sparse crowd on the first play of the second half, when he angled to the left and then found room down the sideline, cutting back toward the middle of the field and then winning a foot-race with the Indianapolis defense for an 86-yard touchdown run.
The Chiefs were in position to take the lead later in the third quarter, but Quinn was picked off by Vontae Davis in the end zone. The defense forced a three-and-out and Kansas City was driving again before settling for Succop's tying 47-yard field goal.
The Chiefs got the ball back again late in the fourth quarter, but Quinn was stuffed on a quarterback sneak on fourth-and-inches from the Indianapolis 27.
That turned the ball over to Luck, who completed an 11-yard pass to T.Y. Hilton to convert a third down, and then found Wayne for two more passes to set up his go-ahead touchdown reception with 4:08 left in the game.
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