Sullivan scored with a one-timer from the left faceoff dot off a pass from Ryan Suter.
"It was just a matter of trying to get open, and [Suter] made a great pass," Sullivan said. "We're in a dogfight for the playoffs, we needed two points, and we knew it."
Nashville had tied it 2-2 on Joel Ward's short-handed goal at 7:05 of the third period. All-Star defenseman Shea Weber gave the Predators their first lead 5:18 later.
"It's not exactly what we wanted, giving up a late goal, but we persevered and got the win," Weber said.
Sullivan's assist on Weber's goal gave the 34-year-old winger 600 career points.
Jack Johnson forced overtime by scoring the Kings' final goal with 1:04 left.
"We're not going to quit just because things don't look good for the playoffs," Johnson said. "We had guys going to the net, and I was just trying to get a shot through and give them a chance to get a rebound."
Los Angeles is far outside the Western playoff race.
"I really like what we have been doing in third periods lately," Kings coach Terry Murray said. "We've been battling, finding ways to win games, to tie up games to get them to an overtime situation, and tonight was another example of that."
Nashville moved up to seventh place in the Western Conference, one point ahead of the idle Edmonton Oilers. Of the Predators' seven remaining games, five are on the road, and all but one of their opponents appears headed for the postseason.
The Predators are 4-0-3 in their last seven games. The last six have been decided by one goal.
"It's taxing our nerves, it's taxing our resiliency, and it's taxing our character a little bit lately," Nashville coach Barry Trotz said. "It has been a rough week. We have played three games in five nights, and they have all been emotionally draining in one way or another."
Brian Boyle staked the Kings to a 1-0 lead 7:00 into the first period. Boyle stripped Weber of the puck behind the net and beat goalie Pekka Rinne on the short side with a wrist shot along the ice.
"It was nice since I had a little bit of nerves coming in, so it was good to help out right away," Boyle said. "I think it deflected off of something and it was able to get up over his pad."
Boyle was recalled Friday from Manchester of the American Hockey League after Jarret Stoll was injured Thursday at Dallas.
Nashville tied it with 5:36 left in the period on < a href="http://sports.espn.go.com/nhl/players/profile?playerId=3578">Cal O'Reilly's first NHL goal.
O'Reilly intercepted Brad Richardson's pass inside the Kings zone and sent the puck to Sullivan at the right dot. Sullivan returned the puck to O'Reilly in the high slot, where his one-timer beat Jonathan Quick on the glove side.
The Kings regained the lead at 8:56 of the second on Matt Greene's first goal in 44 games.
Ward's tying, short-handed goal came at the end of a 2-on-1 off a pass from Martin Erat, who had a career-high four assists.
"If you have the chance to create something, you have to take the chance -- especially when you are down by one and your playoff lives are at stake," Ward said.
Los Angeles C Anze Kopitar recorded his 200th career NHL point. ... Quick has started 26 of Los Angeles' last 30 games. ... Erat has a three-game point streak. He earned his 300th NHL point. ... O'Reilly was recalled by Nashville from Milwaukee (AHL) on Thursday after C David Legwand was injured in Wednesday's practice. ... Nashville swept the season series 4-0.