Brian Willsie and Raitis Ivanans, who totaled seven goals in the Kings' first 80 games, both scored against Vezina Trophy candidate Evgeni Nabokov before Kopitar got his 31st of the season with 5:38 remaining for the go-ahead goal and added an empty-netter with 4 seconds left.
Rookie Erik Ersberg made 32 saves for Los Angeles.
The loss was the Sharks' first that didn't involve overtime or a shootout since Feb. 20, when the New Jersey Devils beat them 3-2 in Newark. San Jose lost to Edmonton on March 16, snapping a franchise-best 11-game winning streak, then lost in overtime four games later at Phoenix on Martin Hanzal's overtime goal.
By dropping their next-to-last game of the regular season, the Sharks won't become the first post-1980 expansion team to put together back-to-back 50-win seasons. Nor will Nabokov get a chance to tie the NHL record of 47 victories set last season by New Jersey's Martin Brodeur. The Sharks also failed to get their first-ever sweep of the four road games against the Kings.
Joe Thornton had a goal and an assist for San Jose and Joe Pavelski got the tying goal on a power play with 16:57 left in the third period.
The Kings, who came in averaging the second-fewest penalty minutes in to the league (11.7), didn't have to play short-handed until Alexander Frolov was sent off for hooking Marc-Edouard Vlasic with 17:33 left in the third period. The Sharks tied it 2-all just 36 seconds later, as Pavelski beat Ersberg high to the stick side from 20 feet for his 19th goal.
Los Angeles' first two goals came on short wrist shots by Willsie and Ivanans that beat Nabokov between the pads and trickled across the goal line. Willsie's third of the season tied the score at 2:14 of the second period after Michael Cammalleri knocked the puck off defenseman Kyle McLaren's stick as he was attempting to clear it from in front of the net. Ivanans' sixth goal put the Kings ahead at 10:29 of the period after Frolov set him up from behind the net.
Thornton opened the scoring just 36 seconds after the opening faceoff, banking the puck off Ersberg and into the net from the left corner for his 29th goal and ninth in seven games. The Sharks have scored the first goal a league-leading 55 times this season (38-12-5) and have done it 18 times during this 18-1-2 surge.
The Kings, who will finish last in their division for the seventh time in franchise history and first time since 1998-99, are tied with Tampa Bay in the race to avoid finishing with the league's worst record. Both teams have 71 points and play their season finale on Saturday, with Los Angeles hosting Anaheim and the Lightning visiting Atlanta. The only time the Kings had the fewest points in the league was 1969-70.