San Jose ended a four-game losing streak, getting two goals from Couture.
Couture produced his 11th and 12th goals of the season, scoring his second one just 27 seconds into the third quarter.
That gave the Sharks a 4-1 advantage, but Dustin Brown delivered his second of the game with 7:12 to go to give the Kings hope and Dwight King scored on a slap shot with 3:20 to play.
Antti Niemi, who faced 37 shots, then settled down to preserve the victory.
The Kings have lost two in a row for the first time in more than a month.
Ukrainian protestors topple Lenin statue [VIDEO]
'SNL': 'Anchorman 2' cast, One Direction sing 'Afternoon Delight' [VIDEO]