Ryan Getzlaf tied it at 2 for the Ducks early in the third period, but Dustin Brown and Kopitar responded with power-play goals to give the Kings a 4-3 victory against their Southern California rival.
"Before I don't know if we were comfortable going into the third period with a one goal-lead, but we kept our composure," Kopitar said. "We got scored on but we didn't back off, we kept on going."
Chris Kunitz scored for Anaheim with 3:40 to play, but Jonathan Quick shut down the Ducks the rest of the way, finishing with 28 saves for his second successive victory against Anaheim. He also backstopped a 4-3 win in January.
Each team finished 3-2-1 against the other.
"This team has been playing great on the road," said Quick, who has been in goal for all of the Kings' recent road success. "We were up by a goal with a couple of minutes left and I don't think they got into our zone."
Getzlaf snapped a shot over Quick during a power play 1:41 into the third to bring the Ducks even. Teemu Selanne also scored on the power play for Anahiem in the second.
The Ducks won the season series between the local rivals in the past three seasons, going 18-7-2 against the Kings since October 2005. But the 2007 Stanley Cup champions have struggled, going 14-18-2 overall since Dec. 1 while dropping out of a playoff position.
"We are not comfortable with where we're at," Ducks coach Randy Carlyle said." We haven't played to a high enough level. We're in a situation where if we don't get points, it's a grim outlook. I think that the difference in the game from our standpoint tonight was their power play and penalty kill won them the hockey game."
It doesn't help that the Ducks begin a 13-day, six-game road trip in Detroit on Friday.
"It will be tough. We're playing some good teams on the road," said Ducks captain Scott Niedermayer, who was in the penalty box for both of the Kings third-period power play goals. "But, maybe that is what the doctor ordered, get out together and us against everybody else. We'll see how it goes."
Stoll gave the Kings a 1-0 lead, tipping in a cross-zone pass from Kyle Calder shortly after Los Angeles killed a power play midway through the first period.
O'Sullivan fought off a check by Todd Marchant to swipe in a rebound of a shot by Greene 9:18 into the second period for a 2-0 lead.
Selanne quickly drew the Ducks within a goal, whisking a shot from the high slot past Quick during a Ducks power play only 1:16 after O'Sullivan had scored.
Anaheim had a chance to tie the game on the power play late in the second, but Quick kicked aside a shot by Bobby Ryan before Selanne was called for an interference call that nullified the advantage.
- Ducks C Samuel Pahlsson remained sidelined with a viral infection and will not travel with the team on its six-game road trip.
- Anaheim leads the series overall, 40-30-19.
- Jonas Hiller had 22 saves for Anaheim.