Kopitar's goal at the end of the second period gave the Kings their first victory in six, while also snapping a six-game winning streak for the Sharks.
Jonathan Quick made 23 saves for his third shut-out of the season, while opposite number Alex Stalock took his shut-out streak to a franchise-record 178 minutes before Kopitar tallied.
Another goalie on form was Marc-Andre Fleury, whose 24-save shut-out lifted the Pittsburgh Peguins to a 3-0 win over the Buffalo Sabres.
Chris Kunitz, Deryk Engelland and Tanner Glass were on the scoresheet for the Penguins.
Eric Staal scored twice and Jordan Staal netted the winning goal in the third as the Carolina Hurricanes rallied from two goals down to beat the Columbus Blue Jackets 3-2.
Elsewhere, the Boston Bruins beat the New York Islanders 6-3, the Colorado Avalanche edged out the Dallas Stars 4-3 and the Edmonton Oilers saw off the Vancouver Canucks 4-2.