Clarkson scored 32 points, including five 3-pointers, and poured in nine straight down the stretch to fuel a decisive 13-0 run.
"I thought our guys just really dug in, stayed the course and stayed focused on the little things," Jazz coach Will Hardy said.
Mitchell, who spent his first five seasons with Utah, also had six assists, five rebounds and three steals. The three-time All-Star was traded to Cleveland in September for a blockbuster package that included Markkanen.
Utah honored Mitchell with a video tribute before tipoff, and Jazz fans gave him a huge ovation during the video and player introductions.
"I was really comfortable," Mitchell said. "It wasn't weird. It was just like playing as I have here for the last five years. The reception was phenomenal, and I'm appreciative of it."
Mitchell eclipsed 40 points for the second time in five games. He set a Cleveland record with 71 points on Jan. 2, most by an NBA player in 17 years, as the Cavaliers rallied to beat the Chicago Bulls in overtime.
According to Elias Sports Bureau, Mitchell is the seventh player in NBA history with 40 or more points in his first road game against a former team. Of those seven players, he is just the second to lose; the other was Nick Van Exel in his first game back in Los Angeles against the Los Angeles Lakers.
Tuesday marked Mitchell's fifth 40-point game of the season, already a career high.
Cavaliers center Jarrett Allen played only seven minutes -- all in the first quarter -- before exiting with an illness. Cleveland forward Isaac Okoro was limited to 17 minutes for the same reason. Both players felt nauseated during the game.
That put a greater onus on Mitchell to make big plays for 48 minutes.
"He was Donovan," Cavaliers coach J.B. Bickerstaff said. "He was trying to will us through it."
Cleveland fell behind by 13 in the second half but chipped away at the deficit, finally erasing it midway through the fourth quarter. Mitchell scored three straight baskets to punctuate an 11-0 run and give Cleveland a 101-94 lead. Utah missed 13 of its first 14 shots in the fourth, opening the door for the Cavaliers to go back in front.
But the Jazz surged ahead 115-107 with 23.1 seconds left after Clarkson scored nine consecutive points over three possessions. His outburst helped Utah overcome a quarter in which Cleveland blocked five shots and the Jazz totaled only five baskets.
"I was just being aggressive in those moments," Clarkson said. "Throughout the game, I saw what they were doing defensively and just started being aggressive in the fourth."
Mitchell kept the Cavaliers' offense afloat in the first half while his teammates struggled to find their shot. He scored or assisted on 10 of Cleveland's 19 baskets before halftime.
When Mitchell cooled off in the third quarter, Utah scored on seven straight possessions to race out to a double-digit lead. Clarkson made three baskets in a row and assisted on another to power a 13-2 spurt that put the Jazz up 78-65.
"He had that look in his eye, and I was just like, damn, it's going to be one of those back-and-forth nights," Mitchell said.
The Associated Press contributed to this report.