Guard Chris Paul scored 26 points including a crucial jumper with just 33.7 seconds left on the clock as the Clippers held on in a frantic finale.
The road victory saw the Clippers improve to 29-16 for the season while Indiana fell to 23-22 after the game at the Bankers Life Fieldhouse.
The Clippers had been plunged into controversy earlier Tuesday day after confirming that Griffin had suffered a broken bone in his hand after throwing a punch at an unidentified person.
The Los Angeles Times reported that Griffin had attacked a member of the Clippers equipment staff during an altercation in Toronto at the weekend.
The incident left Griffin needing surgery to repair a spiral fracture in a metacarpal in his right hand, meaning he faces four to six weeks out.
The 6ft 10in (2.08 meter) power forward now faces an internal investigation after the Clippers announced a sternly worded statement about the incident on Tuesday.
“This conduct has no place in our organization and this incident does not represent who we are as a team,” the Clipper said.
“We are conducting a full investigation with assistance from the NBA. At the conclusion of the investigation, appropriate action will be taken.”
Elsewhere Tuesday, Kevin Durant poured on 44 points as the Oklahoma City Thunder outgunned the New York Knicks in an overtime thriller at Madison Square Garden.
The Thunder squeezed home 128-122 with Durant also contributing 14 rebounds and five assists while draining four three-pointers from seven attempts.
In Toronto, Kyle Lowry scored 29 points as the Raptors defeated the Washington Wizards 106-89 to record their ninth consecutive victory, equalling the franchise record for longest winning streak set in 2002.
DeMar DeRozan chipped in with 17 points for Toronto while Terrence Ross contributed 15 points off the bench, and center Jonas Valanciunas scored 13 points and grabbed 12 rebounds.
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