Dwight King wound up with a hat trick thanks to an empty-net goal, but the Los Angeles win came after the Kings had let a four-goal lead slip away.
Phoenix came back to tie it less than three minutes into the third, but Richards scored unassisted 4:19 into the closing period to give the Kings a 5-4 lead and Matt Fratin added an insurance goal on a tip-in with 7:49 to go.
Both teams came into the game having won two in a row.
King scored two of the goals that gave Los Angeles its four-goal lead and completed the hat trick with 22 seconds remaining in the third period. The Phoenix comeback included a pair of scores from Martin Hanzal.
Hanzal and Shane Doan both scored in the opening three minutes of the third period to wipe out the last of the four-goal Los Angeles advantage.
Mike Smith gave up the first four Los Angeles scores on 15 shots before being replaced by Thomas Greiss.
Jonathan Quick picked up the win despite allowing four goals. He made 24 saves.
Ohio bar shooting arrested, charged with murder
LGBT community has 'bullied the American people': Bachmann