Jamal Crawford had 12 points, and DeAndre Jordan had 16 rebounds despite spraining his right thumb.
George Hill led the Jazz with 18 points and reserve Dante Exum added 10. Joe Johnson, who had 29 points in his Utah debut, was held to seven points on 3-of-9 shooting.
The Clippers took the lead for good in the second quarter on Griffin's fast-break dunk. It was part of a 22-4 run over both halves that pushed their lead to 49-35. Six different Clippers contributed to the spurt, led by Griffin's seven points.
Rivers scored six of the Clippers' first eight points in the fourth, extending their lead to 77-56. The backup was 8 of 11 from the floor.
The Jazz never got a sustained run going in the second half. A pair of free throws by Exum drew them within eight late in the third, the closest they came in the half.
TIP-INS
Jazz: F Gordon Hayward, who averaged 28.3 point, 8.6 rebounds and 4.0 assists against the Clippers last season, sat out his third straight game with a broken finger on his left hand.
Clippers: In the first quarter, Griffin (8,836) surpassed Corey Maggette (8,835) for fourth on the team's career scoring list. ... They won their home opener for the fifth straight year. ... The Clippers have won eight in a row at home, including the final seven games of last season.
BABY GIRL
Griffin's infant daughter attended her first regular-season game in the arms of girlfriend Brynn Cameron.
The baby named Finley is the couple's second together. Their 3-year-old son, Ford, tagged after his mom and new sister carrying a small ball.
It was a family affair, with Griffin's parents in town from Oklahoma and brother Taylor on hand.
UP NEXT
Jazz: Off Monday before playing at San Antonio on Tuesday.
Clippers: Host the Suns on Monday in the season's first back-to-back and second of three consecutive home games.