Cousins played only seven minutes in the first half because of his foul trouble, but he dominated in the second half, when he scored 23 points and grabbed all seven of his rebounds.
The Jazz played without Derrick Favors, who was sidelined with a sore back, and Cousins overpowered Enes Kanter, Andris Biedrins and Rudy Gobert.
Kanter ended up fouling out.
"Obviously he's a major focus for us, [and] we got their bigs in foul trouble," Sacramento coach Mike Malone said. "We caught a little bit of a break with Favors not being healthy, so we wanted to establish DeMarcus in the post [and] feed off him."
Utah used a 16-0 run in the fourth quarter to take a 91-84 lead with less than three minutes remaining.
"I could see all our players their heads were dropping and they were getting down on themselves," Malone said. "[But] that's my job as head coach to make sure I get them not to drop their heads and realize there is still a lot of time left to win this game. And they did that."
Along with Cousins, Isaiah Thomas led the Kings. He finished with 26 points and eight assists.
"I just wanted to be aggressive and make plays," he said. "I saw some openings [and] took what the defense gave me. I tried to get myself going, and that set up everything else. I feel when I'm aggressive to score it opens up my passing lanes."