Johnson scored the only goal of the shootout in the final round as the Kings beat the Islanders 4-3 on Tuesday night.
Johnson faked a forehand and got Yann Danis to move before he slipped in a backhander net to give the Kings their fourth straight win.
"He made a pretty good move," Danis said. "I tried to stack my pads, and somehow, it squeezed through."
The Kings are fighting for a playoff spot, which has made this a fun season for coach Terry Murray. Los Angeles tied for the fewest points in the NHL last season.
"This is the best time I've ever had in coaching, and I started as an assistant in 1982," Murray said. "There are a lot of good young guys on our team. It's a great dressing room, and has great chemistry."
The Islanders got regulation goals from Doug Weight, Tim Jackman and Andy Hilbert. Danis stopped 36 shots, but New York lost its third straight after a four-game winning streak.
Los Angeles opened the scoring just 2:08 into the game. Simmonds, in the left circle, took the puck from teammate Brad Richardson and ripped a quick wrist shot past Danis' short side.
The Islanders tied it at 4:53 when Mike Comrie fed Bill Guerin alone in the slot. Guerin waited for Quick to make a move, and then pushed a soft back-pass to Weight, who immediately beat Quick to the glove side.
Jackman put the Islanders up 2-1 at 4:20 of the second period with a goal from a near impossible angle. Standing along the goal line to Quick's left, Jackman threw a wrist shot toward the net that hit the post and deflected in.
The lead lasted just 67 seconds as Kopitar tied it by finishing a 2-on-1 with Patrick O'Sullivan.
"It's good that we came back a couple of times during the game and in the shootout," Kings captain Dustin Brown said. "These games are huge for us. We can't, and didn't take the Islanders lightly, because a few years ago, we were the Islanders in L.A."
Hilbert got a quick wrist shot off in the low slot and beat Quick to put the Islanders back on top at 9:50.
Frolov made it 3-3 at 6:41 of the third period when he banged in a rebound with 2 seconds left on a Los Angeles power play. Frolov leads the team with 22 goals.
The Islanders didn't register a shot in the period until Sean Bergenheim sent one on Quick with 7:16 left. They then killed off the last 80 seconds of overtime a man down.
"I thought we played well," Guerin said. "Los Angeles is known as being a good forechecking team, and they were, but I think we forechecked them just as well."
Notes: The Islanders have lost three straight to Los Angeles. ... The Kings wrapped up a five-game road trip (4-1), but will make two more visits to the East Coast this season.