San Jose saw its hopes of catching Detroit for the best record in the NHL disappear earlier in the evening when the Red Wings defeated Columbus to clinch the Presidents' Trophy.
The Sharks will settle for the No. 2 seed in the upcoming playoffs.
Los Angeles took a 2-1 lead into the third period only to have the Sharks tie it at 3:03 on a power-play goal from Joe Pavelski.
But Kopitar's wrister beat Evgeni Nabokov and the Kings held on to end a two-game skid. Kopitar added an empty netter with three seconds remaining.
Erik Ersberg made 32 saves for Los Angeles.