Curry followed back-to-back 3-pointers with three free throws, which gave Golden State a 9-4 lead.
The Warriors never looked back, improving their record to 19-10 overall and 11-6 on the road.
Golden State was coming off a 118-115 overtime loss to the L.A. Lakers last Saturday.
"It's a big road win for us," Curry said, "especially the way we finished the game against the Lakers. We were on a mission to come out and play well and keep our road swag [going]. Everybody stepped up, played a physical game and did what we had to do to get the job done."
Along with his scoring, Curry contributed eight rebounds and seven assists against a team that selected his father, Dell, in the first round of the 1986 NBA Draft.
"We obviously know they play well at home and feed off the crowd," Curry said. "And they play hard for 48 minutes. But we do the same. ... We wanted to start off and not let them get any momentum so the crowd would stay out of it as much as possible."
Jazz fans started leaving the arena at the end of the third quarter, when the Warriors owned a 75-58 lead.
"It was a great team effort," coach Mark Jackson said. "We defended ... and we paid attention to detail."
According to Jackson, the key to beating Utah was "making shots."
The Warriors were 9 for 18 on 3-pointers and missed only one of their 20 free throws.
Said Jackson: "Our guys did a great job following the game plan, taking care of the basketball and not allowing them to dominate the paint and the boards."
David Lee finished with 18 points and nine rebounds for Golden State.
"I think, physically, we outcompeted those guys," he said. "Steph, obviously, had an unbelievable game. I just thought we did a very good job of closing them out when we had a lead."