As Celtics players walked off the court through a tunnel toward their locker room, Rondo, dressed in street clothes, greeted them.
"I know he's sad. I'm sad for him," starting guard Avery Bradley said. "Whatever Doc wants me to play, that's what I'm going to do."
Rondo was averaging career highs of 13.7 points and 5.6 rebounds along with 11.1 assists this season.
Still, the Celtics are struggling to remain around .500, finishing the day with a 21-23 record, 2 1/2 games ahead of the Philadelphia 76ers for the eighth and final spot in the Eastern Conference.
The Celtics have no true point guard behind Rondo. Rivers said he didn't know if they would try to add one.
Jason Terry, known for his outside shooting, Courtney Lee, Leandro Barbosa and Bradley are the other guards. Each played at least 25 minutes Sunday with Lee starting in Rondo's place.
"We knew something was up at the beginning, but we didn't know how serious it was," Lee said. "Nobody can fill his shoes. That's what makes him great. So the only thing we can do is work as hard as possible and keep the momentum going from this game and have no letup."
Rondo suffered a dislocated left elbow in the playoffs on May 7, 2011, when he became entangled with Miami's Dwyane Wade.
Boston won 97-81, but Miami won the next two games to clinch the second-round series in five games.