Harden fell one point shy of the franchise record of Hall of Fame guard Calvin Murphy. Harden finished 19 of 25 from the floor, including 7 of 8 from behind the arc, while making 11 of 12 free throws.
Harden added 13 assists in the rout, tallying half of the Rockets' 48 third-quarter points.
With his final basket in the third quarter, Harden eclipsed his previous career high of 53 points. He missed his final four shot attempts in pursuit of the franchise mark before exiting.
Houston (8-3) hit 23 of 39 3-point attempts, with guard Eric Gordon shooting 6 of 11 from deep while adding 20 points. Trevor Ariza tallied 14 points while making 4 of 8 3-pointers. Nene added 13 off the bench.
All five Utah (5-5) starters scored in double figures, but rookie guard Donovan Mitchell paced the Jazz with 17 points off the bench. The Jazz darted to a 10-2 lead before the Rockets stormed back behind Harden, who scored 30 first-half points and made his first 10 shots.
Rudy Gobert added 13 points and five boards for the Jazz while Rodney Hood and Derrick Favors scored 12 apiece. The Jazz dropped their second consecutive game while the Rockets matched their season-best win streak by claiming a third consecutive victory to kick off a three-game homestand.
Houston shot 59.3 percent (48 for 81) from the floor and led by as many as 32 points.
NOTES: With his 29th point, all in the first half, Rockets G James Harden surpassed Hall of Famer Moses Malone for fifth place in franchise history with 11,120 points. Harden opened the game 10-for-10 before missing a floater in the lane late in the second quarter. ... The Jazz surrendered 14 3-point baskets in the first half, more than it allowed in any of the previous nine games. ... The Jazz fell to 0-4 on the road but have a four-game homestand upcoming. Utah has 10 of its first 14 games at home. ... The Rockets made 20-plus 3-pointers for the second time on the season.