The All-Star Game MVP was two rebounds shy of the mark going into the fourth quarter. He grabbed his ninth rebound early in the period and pulled down his 10th with 4:12 to go to become the first Cleveland player with a triple double since LeBron James in the 2009-2010 season.
A young fan wearing a stocking cap raced on the floor and patted Irving on the shoulder midway through the fourth quarter, but was quickly subdued by several policemen and security guards. The fan was wrestled to the floor and led away in handcuffs.
Cleveland dominated the second half to earn the win and gain ground in the race for the final playoff spot in the Eastern Conference.
Luol Deng scored 16 points as the Cavaliers placed five players in double figures.
Gordon Hayward scored 18 points for Utah, which began a six-game road trip.
Irving scored 10 points in the third quarter as the Cavaliers turned a three-point deficit into a 63-51 lead. The margin reached 23 late in the fourth quarter.
Irving played 45 minutes Tuesday against Toronto and 43 the following night in Oklahoma City. On what must have seemed like a short night, he played 38 minutes Friday.
Cleveland's long injury list gained a player when Anthony Bennett, the top pick in the draft last year, missed the game with a sore right knee. Center Anderson Varejao (sore back) and guards Dion Waiters (hyperextended left knee) and C.J. Miles (sprained left ankle) were also sidelined.
The Jazz have lost nine of 11 on the road.
The Jazz missed one 3-pointer in seven attempts in the first quarter and built a nine-point lead. Utah maintained the lead until early in the third quarter when Irving took over. He hit a 3-pointer, scored on a twisting drive to the basket, knocked down two jumpers and hit two free throws. Irving also assisted on Alonzo Gee's buzzer-beating jumper from the corner that gave Cleveland a 69-60 lead.
Richard Jefferson scored 13 points and Alec Burks added 11 for Utah.
Rookie Trey Burke, a native of Columbus, Ohio, played his first NBA game in his home state and was held to four points on 2-of-9 shooting. The University of Michigan product leads all rookies in free-throw percentage and ranks second in assists.
Tristan Thompson had 18 points and 14 rebounds for Cleveland, and Spencer Hawes added 13 points and 16 rebounds.
Hawes was given a technical foul in the third quarter, his second in five games since joining the team last week.