Serge Ibaka of the Thunder and Matt Barnes of the Clippers were ejected, and Griffin received a technical foul for his part in the altercation with 6 seconds left in the second quarter.
Kevin Durant had 33 points, making 15 of 17 free throws, and 10 assists to lead the Thunder, who had won four straight. Russell Westbrook added 19 points and 10 assists.
Jamal Crawford scored 20 points off the bench for the Clippers, who had six players in double figures while winning their third in a row.
The fireworks began when Griffin was trying to put the ball up and his arms and Ibaka's became entangled. Barnes shoved Ibaka hard in the chest. Griffin kept trying to get at Ibaka as players from both teams formed a scrum.
Ibaka cocked his right fist, but didn't take a swing as Barnes smiled at him.
Durant made the free throw on Griffin's technical and Westbrook hit a 3-pointer to give the Thunder a 62-53 lead at the break.
The exchange certainly riled up the Clippers.
Down by 10 early in the third, they rallied with a 14-2 spurt to go up 74-71, their first lead since the opening quarter.
He stole the ball and dribbled down the court, driving in for a layup. After a timeout, Durant lost the ball and Griffin scored again while getting fouled. He made the free throw.
During the Clippers' run, Ryan Gomes, Nick Collison and Thabo Sefolosha all picked up their fourth fouls. Gomes started the second half in place of Ibaka.
Los Angeles shot 7 of 8 on free throws late in the period to take an 83-78 lead into the fourth.
The Clippers forced turnovers and contested shots in the fourth when they weren't scoring. They opened with an 8-0 run to take a 91-78 lead on the strength of their bench shooting.
The Thunder couldn't get a sustained run together while turning the ball over against a team they swept in three games last season.
After his ejection, Barnes used a derogatory term in a profane tweet that began "I love my teammates like family," but went on to seemingly suggest he was tired of backing them up because his actions "cost me money."
The Thunder led by 14 points in the first half before the Clippers closed to 62-53 at the break. Durant, Ibaka and Westbrook all reached double figures in the half, when the Clippers were all in single digits.
Ibaka had 13 points on perfect shooting before his ejection. He hit all six of his shots from the field and was 1 of 1 from the line. Barnes was scoreless in just under 8 minutes.
NOTES: Paul became the first player in Clippers history to have at least 10 points and 10 assists in each of the first nine games of the season. The last player to do so was Magic Johnson of the Lakers in the first 11 games of the 1990-91 season. ... The Thunder was the league's second-best road team last season. ... The Thunder will play on back-to-back nights for the first time this season when they visit Golden State on Thursday. ... Thunder C Kendrick Perkins wasn't with the team after his grandfather died Monday. He was raised by his grandparents in Beaumont, Texas. Clippers coach Doc Rivers, who coached Perkins in Boston, talked to him Tuesday.