Richards, returning to the city where he had played his entire career before an off-season trade sent him to LA, found Johnson in the high slot and Johnson put the puck past Flyers goalie Ilya Bryzgalov 1:39 into the extra session to give Philadelphia its first loss of the season.
The Flyers, who had started the season with three victories, also were victimized by another former teammate, Justin Williams, who scored twice in regulation for the Kings.
Jonathan Quick made 32 saves for Los Angeles, which stopped a two-game losing streak.
The Flyers got goals from Danny Briere and Matt Carle.
Bryzgalov stopped 23 shots.