The Flyers (3-0-1) were trying to start a season 4-0 for the first time since 1995.
The road-weary Kings (2-1-1) began the season in Europe and still haven't played at home. In addition, many of their defensemen - including Johnson (30 minutes, 57 seconds) and Willie Mitchell (27:15) - logged extra minutes because of Doughty's early upper-body injury.
"There were a lot of excuses we could have used tonight, but you see the type of character we have in here," said Richards, who played 20:58 and won 13 of 20 (65 percent) face-offs. "We dug down and found a way to win."
Richards said the game-winner was set up by "a broken play." He skated from the point toward the high slot to set up the goal.
"I was just looking for a lane to shoot, and the D was stepping up and I wanted to try to shoot through him," Richards said. "I saw J.J. at the last minute and I just threw it to him."
Johnson, at the left of the net, tapped it past Ilya Bryzgalov. Johnson also had a four-on-three overtime winner against the Rangers in Stockholm.
Trailing 2-1, Flyers defenseman Matt Carle knotted the score on a power-play goal with 6:31 left. It was Carle's first goal since Dec. 5, 2010.
Carle took a crafty pass from James van Riemsdyk and beat an out-of-position Quick from deep inside the left circle.
Just 54 seconds earlier, Los Angeles had taken a 2-1 lead on Williams' second goal of the night. Williams scored on his own rebound.
With 19.9 seconds left in regulation, van Riemsdyk was called for tripping, and the penalty carried into overtime. It proved costly.
Penalties "away from the goal are usually the ones that come back to bite you - and it did," said defenseman Chris Pronger, whose team outshot the Kings, 34-23, in regulation.
The teams traded power-play goals in an interesting first period, one in which Danny Briere scored after a slick goalmouth feed from Jaromir Jagr, and Williams countered for the Kings.
Just 24 seconds into the game, the Flyers' Zac Rinaldo leveled Doughty with an open-ice hit. Doughty, who signed an eight-year, $56 million contract late last month, returned for one shift and then went to the locker room; the L.A. Times said it was not a head injury, but that Doughty would miss seven to 10 days.
About 11 minutes after Rinaldo's hit, perhaps as retribution, Pronger was slashed by the Kings' Dustin Brown on his right hand - the one that he had surgically repaired in March. Pronger continued taking a regular shift.
Richards, booed loudly every time he carried the puck, sent Jakub Voracek to the ice with an open-ice check with 12:40 left in the second period. Wayne Simmonds, one of the players acquired in the blockbuster deal that sent Richards to the Kings, skated to Voracek's aid. He and Richards shouted at each other, but no gloves were dropped.
Richards was jawing with the Flyers bench during the game; afterward, he said his ire was directed at Rinaldo.
"Oh, the guy that didn't play very much," Richards said. "The guy that hit Doughty. Nobody likes that. It's tough on the ice when you just go out there and try to run around; it's tough to play against those players, but at the same time, we did a good job."
Pronger said he wasn't surprised there was a Richards-created buzz in the crowd all night.
"He spent six years here and was the face of the franchise," Pronger said. "I think the fans showed their appreciation - and then they got over it quickly."
Pronger was referring to a standing ovation Richards received early in the game, followed by loud boos whenever he touched the puck.
Contact staff writer Sam Carchidi at firstname.lastname@example.org or on Twitter @BroadStBull.