Gardner Minshew took over from there, closing out a 31-20 victory for the Colts.
Richardson rushed for two touchdowns and Minshew threw for 171 yards and a TD in the first win for coach Shane Steichen, who was hired in February after spending the last two seasons as Philadelphia's offensive coordinator.
"There are certain things that we don't do with Gardner that we do with Anthony," Steichen said. "But as far as the passing game, I thought he was efficient and operated at a high level. Just a great player, a great veteran guy. He prepares like he's the starter but also does a heck of a job helping Anthony out throughout the week."
The Colts led by 18 at halftime and managed only a field goal in the second half. But their dominant start was enough to snap an eight-game skid dating to last season. It was their first win since Nov. 13 at Las Vegas.
"I told him at halftime, 'I'm going to hold it down for you,'" Minshew said of a conversation with Richardson. "Everybody follows him, he's the leader of our team, and I was happy to be able to fill in and help when I could, but I'm going to continue to help him and support him as much as I can."
Richardson gave the Colts (1-1) an early lead when he dashed 18 yards untouched on their first drive.
The rookie QB, who was selected by Indy with the No. 4 pick in the NFL draft, hit the back of his head when he was pushed down at the end of the score, but he popped right up to celebrate. He remained in the game for two more possessions before going to the medical tent and then to the locker room.
Steichen said he was injured on that second touchdown run and that he didn't report the injury to the staff until after he played two more drives.
"It looked like a fluke thing on that one, but hopefully it doesn't continue to happen," Steichen said.
Defensive tackle DeForest Buckner presented Steichen with a game ball to commemorate his first NFL win.
"He's trying to play it all cool, but then when (he) gave him the team ball, his voice was cracking so he liked that," Minshew said.
Stroud, the second overall pick in the draft, threw for 384 yards and two touchdowns. But he was sacked six times and hit nine other times playing behind a line filled with backups. The Texans (0-2) lost four starters on the line during camp and were also without star left tackle Laremy Tunsil on Sunday because of a knee injury.
"Guys are still going out trying to do the best they can do," coach DeMeco Ryans said. "Doing the job that we ask them to do. Of course, we would like it better. Don't want to see the quarterback get hit. But I thought the offense took a move in the right direction today, got better overall."
The Texans closed to 31-20 on a 23-yard touchdown reception by rookie Tank Dell with about 10 minutes to go. It was the first career touchdown for the third-round pick from the University of Houston.
Houston got a stop after that but wouldn't get any closer after missing a long field goal on the next possession.
Stroud threw his first career touchdown pass when he connected with Nico Collins for a 12-yard score that cut the lead to 14-7 with about three minutes left in the first quarter. Collins finished with seven receptions for a career-high 146 yards.
Minshew took over for Richardson in the second quarter and orchestrated a 76-yard drive capped by an 11-yard scoring run by Zack Moss that made it 21-7.
The Colts pushed it to 28-10 when Kylen Granson scored on a 4-yard pass just before halftime.
Richardson is the third QB in NFL history with three rushing touchdowns in his first two games, joining Daunte Culpepper and Jack Thompson. Richardson, who is 21 years and 118 days old, is also the youngest quarterback with two rushing touchdowns in a game.
Josh Jones filled in for Tunsil and had a holding penalty in the fourth quarter that negated a touchdown by Dell. The Texans settled for a field goal on that drive to trim their deficit to 31-13.
Houston wide receiver John Metchie III made his NFL debut after missing all last season undergoing treatment for leukemia. Metchie, who sat out last week with a hamstring injury, had one reception for 17 yards.
Metchie was disappointed with the loss but knows that his return to the field means a lot to many people who followed his fight.
"(It) was not to be satisfied with just coming back, but strive to be the best in coming back," he said. "And that was kind of my mindset that got me through that period of time. Not just wanting to ... make it out. I wanted to thrive, not just survive."
Colts: Visit Baltimore next Sunday.
Texans: Visit the Jaguars next Sunday.
AP NFL: https://apnews.com/hub/nfl