Joe Mixon rushed for 102 yards and caught a TD pass from Stroud after the veteran missed the Texans' previous three games with an ankle injury.
"He's got a great feel for the game and has great vision," Stroud said.
Stefon Diggs had six catches for 77 yards and a score for the AFC South-leading Texans (5-1), and Tank Dell had seven receptions for 57 yards and a TD.
Dameon Pierce ripped off a 54-yard touchdown run in the fourth quarter and finished with 76 yards on eight carries. Will Anderson Jr. recorded a career-best three sacks for Houston, which has won three straight.
"You've got to keep chopping and going and keep getting better," Anderson said. "I have to give credit to the defensive linemen. Those guys are tremendous in what they do. I love those guys."
The Patriots (1-5) got a boost from Maye, the No. 3 overall pick in the NFL draft. Taking over for veteran Jacoby Brissett, he finished 20 of 33 for 243 yards and three touchdowns. He also ran for 38 yards but was intercepted twice and sacked four times, one resulting in a lost fumble deep in New England territory that led to a Houston touchdown.
After sacking No. 1 overall pick Caleb Williams seven times in a win over Chicago last week, Houston wasted little time in making Maye feel uncomfortable. The Patriots went three-and-out on their first possession, and on their next series, Maye overthrew his intended target and was intercepted by Calen Bullock — one of four turnovers by New England, which has lost five straight and fell to 0-3 at home this season.
"The thing about this team is taking the ball away," Anderson said. "If you get a lot of turnovers, you win the game."
Maye threw a 40-yard touchdown pass to Kayshaun Boutte with 11 seconds left in the first half. The rookie equaled Brissett's TD output through five games (two) when he found tight end Hunter Henry in the left flat for a 6-yard score that cut into Houston's lead to 27-14 in the third quarter.
Maye gave the Patriots fans who stuck around with one more reason to cheer when he connected with DeMario Douglas for a 35-yard TD in the fourth quarter. Douglas had six catches for 92 yards.
"I thought he showed a lot of poise. I thought he went out there and controlled the huddle, got those guys out of the huddle and once again, made some plays. It's definitely encouraging," Patriots coach Jerod Mayo said of Maye. "From a team-wide perspective right now, we let him down. It was his first game, and I feel like I let him down. I'm sure all the coaches feel like we let everyone down. We've just got to be better.
Both teams took the field without key players on offense. The Texans placed top receiver Nico Collins on injured reserve with a hamstring injury, while the Patriots were minus running back Rhamondre Stevenson, who has a foot injury.
Stroud finished 20 of 31 for 192 yards. He was intercepted once.
"We knew that we needed everyone to step up. Once everyone took on that challenge, we started rolling," Stroud said. "We couldn't feel bad about Nico not being there. We had to pick up the slack."
Houston won at New England for the first time. The Texans had been 0-7 at Gillette Stadium, including playoffs.
Mixon and Pierce led a Houston rushing attack that finished with 192 yards.
"Those guys bring playmaking ability," Houston coach DeMeco Ryans said. "Very excited to see those guys continue to do more for us, but very pleased with the run game and how they got after it."
One week after committing 12 penalties for 105 yards in a loss to visiting Miami, the Patriots were flagged on the game's opening kickoff for an illegal formation. That was just the beginning, with New England getting flagged for pass interference on third down to set up each of Houston's first-quarter touchdowns.
All told, New England was penalized nine times for 50 yards.
"It's the same story over and over again," Mayo said.
The Patriots reached a season high in scoring behind Maye, but they also gave up their most points this season.
Bryce Baringer's 54-yard punt in the second quarter extended the Patriots punter's NFL-leading streak to 23 games with a kick of at least 50 yards.