Tyler Bozak also scored and Jordan Binnington made 26 saves for the Blues, who are locked into the fourth seed in the West Division and will play Colorado or Vegas in the first round of the playoffs.
"We want to keep building," Bozak said. "We want to limit our turnovers. Obviously, create a lot of O-zone time. I still think we could get a lot more shots on net, make more plays in the offensive zone that way, but we'll keep building here moving into playoffs."
Faulk and David Perron sprung a 2-on-1 rush after Ryan O'Reilly won the opening faceoff in overtime, and Faulk won it with a wrist shot for his seventh goal.
"It's good to see him get a goal," Blues coach Craig Berube said. "He gets a lot of opportunities now night in and night out to score. Could easily have five, six more goals this year probably, but just a warrior. He just keeps going. He doesn't stop. Nothing bothers him."
Alex Iafallo scored, Calvin Petersen made 17 saves and the Kings wrapped up their home schedule with four straight losses.
"Offensively, I thought for a team that really has to work hard to score, I thought we got cute," Kings coach Todd McLellan said. "We wanted to make plays when we should have been shooting the puck and it probably cost us the game."
After two periods of tepid hockey, which Berube said had no emotion, the Blues and Kings finally started to generate offense in the third.
Bozak went five-hole with a backhand to give St. Louis the lead 1:25 into the third.
Jaden Schwartz set up Bozak's fifth goal, and Faulk had the secondary assist, his fifth in the past five games. It was also Faulk's 200th career assist.
Iafallo responded with a one-timer from the left circle to tie it at 7:35.
Gabriel Vilardi extended his point streak to four games with the assist. He has three goals and two assists in that span, with the 2017 first-round draft pick closing out an erratic first full season in the NHL on a positive note.
"I'm feeling comfortable right now," Vilardi said. "I'm playing with two good linemates that both work hard, and I'm able to get the puck so I can make more plays. We're creating right now offensively."
Drew Doughty nearly won it in regulation for Los Angeles, hitting the post on a power play with 1:09 remaining, but Binnington was able to fall on the loose puck before the Kings could react.
St. Louis finished 15-9-4 on the road this season.
"We've had success since I've been here playing on the road," Bozak said. "I'm not sure what it is. Maybe we play a little simpler game which better suits us on the road, but home-ice advantage, you want to play your best at home in front of your fans. And we're gonna have to be better than what we were this year at home, and we've got two games to kind of improve on that to finish the year and hopefully end the year off on a good note, head into playoffs confident."
WORK TO DO
Los Angeles was 9-14-5 at home and lost 11 of its final 14 games at Staples Center after working its way into playoff contention in March. The Kings never won more than two in a row on home ice.
"Our younger players that have been in the league now for a year or two must show improvement," McLellan said. "They have to begin to carry more of the load, offensively, defensively, game management. It's a must that they graduate and move on to the next level, break through."
Blues defenseman Torey Krug returned after missing five games because of an upper-body injury, and Berube expects several other injured players could get at least one game in before the playoffs start.
That group includes forwards Vladimir Tarasenko and Sammy Blais, and defenseman Vince Dunn.
Dunn has missed nine games because of an upper-body injury, Blais four games due to an upper-body injury, and Tarasenko four games with a lower-body injury.
ABUNDANCE OF CAUTION
Blues defenseman Jake Walman did not play after testing positive for COVID-19. General manager Doug Armstrong believes it was a false positive because Walman has been vaccinated and a second test came back negative.
With Niko Mikkola also unavailable because of an upper-body injury, Steven Santini replaced Walman in the lineup.
Blues: Host the Minnesota Wild on Wednesday night.
Kings: At the Colorado Avalanche on Wednesday night.
Faulk's OT goal wins it for Blues vs. the Kings
Justin Faulk nets the game-winning goal in overtime as the Blues edge out the Kings 2-1.