The Texans have won 11 games under three head coaches over the last three years, not including an interim coach. They have 10 wins this season and their first playoff berth since 2019.
Houston defeated Indianapolis 23-19 in a win-or-go home game Saturday night.
The Texans (10-7) can win the AFC South if the Titans defeat the Jaguars on Sunday. The Colts fell to 9-8.
CJ Stroud, the front-runner for Offensive Rookie of the Year, threw two touchdown passes and had a crucial drive in the fourth quarter that led the Texans to victory. He completed 20 of 26 for 264 yards.
He threw a 75-yard touchdown to Nico Collins on the team's first snap and a 1-yard touchdown to Andrew Beck in the final two minutes of the first half.
Collins caught nine passes for 195 yards, while six other players caught 11 passes for 69 yards. The Texans were missing their third, fourth and fifth leading receivers.
Stroud led the Texans with a 12-play, 73-yard scoring drive, with Devin Singletary scoring on a 3-yard run with 6:20 left. They overcame a second-and-20 after a sack of Stroud by Ronnie Harrison Jr. and a first-and-20 after a holding penalty on Beck.
However, kicker Ka'imi Fairbairn missed the extra point wide left and put the Texas defense in trouble.
The Colts reached the Houston 15 with 1:06 left and faced a four-and-one. Instead of running Jonathan Taylor, who had 30 carries for 188 yards and a touchdown, Shane Steichen opted to put the ball in the hands of Gardner Minshew. Running back Tyler Goodson was open, but the throw was slightly behind him and he dropped it.
The Texans ran three times and secured the punt, leaving one second on the clock. The Colts only managed 7 yards on the free kick before Houston recovered a lateral attempt from Michael Pittman.
Taylor briefly left the game at the end of the third quarter with heel and ankle injuries but returned. He kept the Colts in the game, with Minshew struggling to go 13 of 24 for 141 yards.