Richardson redirected a shot from the point from Jack Johnson past Ducks goalie Johan Hiller and snapped a 3-3 tie with 2:20 to play in regulation.
Johnson had two assists in the game. The Kings also got goals from Randy Jones, Justin Williams and Scott Parse.
Perry put the Ducks on the board after Los Angeles had jumped to a 2-0 lead on goals by Parse and Williams.
Jones's slapshot at 6:40 of the second period put the Kings up 3-1, but Anaheim responded with a power-play goal by Whitney in the second period and a goal from Marchant in the third that tied the score.