He went directly to a hospital.
The forward's right elbow was oddly swollen, prompting Clippers trainer Jason Powell to advise him to get it checked out.
Doctors determined Griffin had a staph infection that would require surgery that will sideline him anywhere from 10 days to six weeks. He will have the procedure Monday in Los Angeles to remove the infection and could miss from four to 20 games.
"It's the unknown," Rivers said. "If it's a sprained ankle, you know [it's] 10 days. With infections, you just have to be very careful. You cannot mess around with them."
The Clippers expect to have a better idea of Griffin's timetable after his surgery. He will miss the All-Star game Feb. 15 in New York, with Portland's Damian Lillard taking his place.
Clippers center DeAndre Jordan said Griffin was struggling with news of his diagnosis.
"He's definitely upset he couldn't be there for us," Jordan said. "But, you have to look out for your health. That's the most important thing. We're all going to back him up and hopefully he has a speedy recovery and we get back to business with a full group of guys."
The recovery time for staph infections can vary widely. Forward Paul Pierce missed only about a week with an infection in his left elbow during the 2006-07 season but forward Grant Hill needed intravenous antibiotics for six months and missed the entire 2003-04 season after developing an infection in his ankle in March 2003.
The timing of Griffin's injury was somewhat fortuitous given that it came shortly before the weeklong All-Star break that starts Thursday, meaning he will miss fewer games than he would have had he needed surgery at another point in the season.
Griffin is averaging 22.5 points, 7.5 rebounds and 5.1 assists per game while shooting 50.1%. He did not speak with reporters Sunday.
Clippers point guard Chris Paul used the word "tough" 10 times during a two-minute interview that revolved mostly around how Griffin's absence would affect the team.
"It's tough," Paul said. "It's a tough loss. It's tough losing Blake."
Spencer Hawes replaced Griffin in the starting lineup against Oklahoma City and scored 17 points, but Rivers said he would vary his lineup depending on matchups.
When Glen Davis left the game against the Thunder in the second quarter because of back spasms, it left the Clippers with three available big men in Jordan, Hawes and seldom-used reserve Ekpe Udoh.
Rivers said he would probably sign another player to a 10-day contract but noted it wouldn't necessarily be a big man because shooting guard J.J. Redick continues to be sidelined by back spasms and guard Austin Rivers has a sore foot.
"We may even bring a guard in and just go small," said Doc Rivers, adding that the new player probably would not join the Clippers before their game against Dallas on Monday. "We just need another player for sure."
Griffin has experienced issues with his right elbow previously. In 2012, he suffered a burst bursa sac and the next year he had the bursa sac drained after his elbow became inflamed. He had his elbow drained in New York last week while dealing with bursitis, but the staph infection caught the Clippers by surprise.
"We were kind of joking around with him when he didn't have the pad on it, how big it was getting, and obviously I didn't think that was the cause," Hawes said. "I'm just glad they caught it and he got it taken care of."