The Warriors snapped an 0-8 start away from home to improve to 8-9 on the season. Golden State has won five of its last seven games following a 3-7 start.
"It didn't sit well with us, our 0-8 record," Curry said. "We had to scratch and claw to get this one, so to get that monkey off your back and feel good about yourselves outside of the Chase Center, it matters."
Thompson had a vintage performance, hitting a sizzling 10 of 13 3-pointers and went 14 of 23 from the field.
"It's just something for me to build on," Thompson said. "I was really wanting a game like this, so it felt great for me to just let that thing fly."
Longtime teammate Draymond Green said of Thompson: "He's the most competitive guy that I've ever played with, and the most important thing to him has always been winning."
Curry struggled early but finished 11 of 20 from the field, including 7 of 14 from 3. He also had a season-high 15 assists, just one shy of his career high.
Curry hit a big 3 just inside of two minutes remaining that doubled the Warriors' lead to 121-115. Curry made another with less than 40 seconds remaining to put Golden State up by nine, and he gestured to the crowd like he put the game to sleep.
"I surprised myself with that shot," Curry said. "Me and Draymond put our hands above our heads like we really won something, kind of making fun of ourselves for our slow start."
Andrew Wiggins hit 6 of 11 3s and scored 22 points for Golden State.
Golden State jumped to an 11-0 lead from the tip and led 40-28 after 20 first-quarter points from Thompson. Houston quickly answered, opening up the second quarter on a 17-2 run to build a small lead.
The Rockets led 65-61 at halftime.
"We played a great team, we got down big and fought back," Rockets coach Stephen Silas said. "We took the lead, we were neck-in-neck down the stretch, and they had to make some great shots to beat us.
"We have to learn some discipline, but this is a team that's been together for years and won championships and all that. We should definitely be proud. It's disappointing that we didn't get the win, for sure, but nobody in that locker room should have their head down."
The loss dropped Houston to 3-14, the worst record in the Western Conference.
Warriors: Golden State received two delay of game warnings in the first half, resulting in a technical. Houston later received the same technical in the second half. ... Jordan Poole had 10 points off the bench.
Rockets: Porter returned after a one-game absence with left lower back soreness. ... Jalen Green committed an uncommon turnover near the end of the third quarter, walking the ball slowly up the court following a Warriors basket and receiving a 10-second backcourt violation.
After Thompson promised on Instagram last week that "the floodgates will open" following early struggles, he said Sunday, "I can confidently say they're open now."
Warriors coach Steve Kerr said he hoped Thompson's "ffollowers are happy with Klay promising that. I hope there's a lot of good interaction between followers and Klay and everybody can be happy."
RAPPERS IN ATTENDANCE
Hip-hop artist 50 Cent sat courtside, as did Houston-native rapper Tobe Nwigwe, who was recently received a Grammy nomination for Best New Artist.
GREEN AND GOLD
The Rockets wore throwback white jerseys with gold lettering and green trim as a nod to the original San Diego Rockets.
Warriors: Visit the Pelicans in New Orleans on Monday night.
Rockets: Have several days off until hosting the Hawks on Friday night.
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