Slava Voynov scored for the Kings, who have won four of five.
Quick turned aside Craig Smith, Filip Forsbeg and Matt Cullen in the shootout. He has denied all eight shots he has faced in shootouts this season, winning all three he has participated in.
"It is kind of a crapshoot to be honest," Quick said of the shootout. "We're fortunate, so far it has gone our way this year so far. At the end of the day we got two points in a tough building against a conference opponent."
Kopitar skated down the right side and elevated a backhand over the left shoulder of Rinne.
"I saw Jeff and Mike try and shoot, so I figured I would mix it up a little bit," Kopitar said. "I guess I was fortunate that my move worked."
Cullen scored his first goal with the Predators at 10:33 of the opening period for the early lead. Nashville finished a five-game homestand 3-2.
After Quick made a diving save to deny Nashville's Gabriel Bourque from the slot, Cullen was able to corral the puck in traffic, move to open space and beat a fallen Quick with a wrist shot from the right side.
"I just want to contribute," Cullen said. "They haven't been coming easy, so it was nice to see a good ending."
Nashville has scored first in three consecutive games.
"We just couldn't find an answer for Jonathan Quick," Nashville coach Barry Trotz said. "He made some incredible saves in the second period. I think if he didn't have a night like he did, we probably would have lit them up."
Los Angeles tied it just before the first intermission when Voynov scored at 19:18.
At the left faceoff dot, Richards froze Rinne and then fired a cross-ice pass to Voynov in the right circle, where he was able to beat a diving Rinne for his second goal of the season.
Voynov has scored in consecutive games.
With just over six minutes remaining in the third, Rinne denied Richards on a short-handed breakaway attempt to keep the game tied at 1.
Nashville had the only power-play opportunity of overtime, but Quick denied Nashville's two shots, both by Patric Hornqvist.
"It takes a lot of courage to kill a 4-on-3," Los Angeles coach Darryl Sutter said. "Because generally you have to get in front of a puck, a hard shot somewhere."
Rinne played his 300th career NHL game. ... Smith has points in three straight games. . The Kings denied Nashville on all four power play attempts in the game. They have not allowed a power-play goal against in their last three.
Copyright by STATS LLC and The Associated Press
NASHVILLE, TN - OCTOBER 17: Kevin Klein #8 of the Nashville Predators laces his skates prior to a game against...