Leon Powe took Perkins' place in the starting lineup. It was the first playoff start of Powe's career.
"I know I'm not ready to go," Perkins said in the visitor's locker room at Staples Center, wearing street clothes while the rest of his teammates got in uniform for the potential clincher. "If we have a Game 6, I'm sure I'll be ready to go. But I hope we close it out."
The Celtics took a 3-1 lead into the fifth game of the best-of-seven series but lost 103-98. Game 6 will be in Boston on Tuesday night.
Celtics coach Doc Rivers said it was doubtful Perkins would be able to play then.
"Obviously, it's not a great injury to have," he said.
A victory on Sunday would have ended the season with Perkins on the bench.
"I'll be cool with that," he said.
Perkins dislocated his left shoulder in 2006 and missed three games, but needed surgery after the season to repair it.
Asked on Sunday if he was always in pain or just when he moved it, Perkins said, "Any moment, it's painful."
"It seemed like it was not getting better that quickly," he said.
Perkins' absence seemed assured when he left Thursday's game clutching his left shoulder after slamming into Lakers forward Lamar Odom early in the third quarter. When the Celtics returned to practice Saturday, Perkins seemed unable to move his left arm, dribbling and shooting with his right hand only.
But the Celtics center expressed optimism and said there's a "strong possibility" he would play.
"He called me this morning and he said there's just no chance of it," Rivers said. "We're prepared for it. We can go a lot of ways."
Rivers said Perkins absence limited forward Kevin Garnett's ability to roam around on defense and force the NBA's defensive player of the year into the lane more. He ended up in foul trouble.
"Kendrick really plugs up the middle for us," said P.J. Brown, a backup center who played more in Perkins' absence. "He's one of our best defensive players, a great rebounder and a big physical presence that can intimidate when guys come to the basket. So we're going to miss that."
Powe has been used sparingly in the playoffs, but he scored 21 points in 15 minutes in Boston's victory over Los Angeles in Game 2. He only played five minutes, however, and didn't score.
Brown has averaged 2.7 points and 2.3 rebounds in 13 minutes in the playoffs this year after sitting out most of the season as a semiretired free agent and joining the Celtics in March. He played 25 minutes and had four points and three rebounds.
"When you've been playing this long, those are the type of injuries that kind of hop up," Brown said, "you know, little nagging, freakish things that both teams have to deal with."