Michael Cammalleri and Tom Kostopoulos scored first-period goals and Dan Cloutier made 23 saves for the Kings. Richard Park and Alexei Yashin scored for the Islanders, who are off to a franchise-worst 0-3 start.
The Kings snapped a 2-2 tie with 3 seconds left on Yashin's hooking penalty. Oleg Tverdovsky threw the puck into the slot from the right circle and Sopel one-timed it over goalie Rick DiPietro's glove with heavy traffic in front of the crease.
The goal was Sopel's second in three games, matching his entire total in 68 games last season with the Islanders and Kings. Los Angeles acquired him in a trade with New York on March 8, and he played in 11 games for the Kings before undergoing surgery on his right knee in May.
The Islanders nearly tied it with 2 minutes to play, but former Kings left wing Jason Blake hit the right post from short range. The play was reviewed and the Kings escaped with the two points.
Cammalleri tied it 1-all at 9:47 with a 30-foot wrist shot after skating across the blue line with a pass from Derek Armstrong. Kostopoulos then converted a rebound of Sopel's wrist shot at 17:36 after defenceman Tom Poti's backhanded clearing pass didn't make it out of the Islanders' zone.
Yashin tied it 38 seconds before intermission with his first goal of the season, a 40-foot slap shot that broke off Cloutier's glove and dribbled across the goal line with the teams skating 4-on-4.
The Islanders converted their first shot on net 76 seconds after the opening faceoff, as Park's one-timer from the left circle found a crack of daylight between Cloutier's arm and the left post after Shawn Bates centred the puck out from behind the net.
Notes: Kings coach Marc Crawford, who won the Jack Adams Trophy as the NHL's top coach in 1994-95 with the Quebec Nordiques, tied Adams for 15th place on the career victory list with his 413th. ... DiPietro made 35 saves. ... The Islanders were 0-for-7 on the power play and the Kings were 2-for-12 ... The teams had not faced each other since Feb. 16, 2004, a 1-1 tie at Nassau Coliseum. ... The Islanders conclude their season-opening four-game road trip Wednesday night at Anaheim.