Zack Kassian scored late in the second for the Oilers, who have lost six straight games and eight in a row at Staples Center going back to a 3-0 victory on Nov. 3, 2011. Cam Talbot stopped 34 shots.
The Kings, coming off consecutive 2-1 wins over Nashville and Calgary, increased their Pacific Division lead to four points over Anaheim.
The Kings are 16-1-2 in their last 19 meetings with the Oilers, who have the fewest points in the NHL.
Carter tied it with 14:33 left in the third, poking the puck past a sprawling Talbot for his 15th goal after the goalie stopped Tyler Toffoli's 25-foot wrist shot and his ensuing rebound attempt.
King added his sixth of the season and first game-winner with 5:50 to play, redirecting a long wrist shot by Carter.
The game was still scoreless when Quick stopped Jordan Eberle's 25-foot wrist shot from the right circle with his outstretched right leg, then dove toward the left post to make a stop on rookie Connor McDavid's rebound attempt with about 5 1/2 minutes left in the second period.
Kassian put the Oilers in front at 18:55, using defenceman Luke Schenn as a screen and beating Quick to the stick side with a 25-foot wrist shot from the slot after getting the puck from Taylor Hall at the top of the left circle.
NOTES: Four days before the NHL trade deadline, Edmonton RW Teddy Purcell was a healthy scratch for the first time this season. That leaves Hall and C Mark Letestu as the only Oilers to play in all 62 games. . Quick is 18-1-5 in 23 games against Edmonton. . The teams conclude their season series March 26 at Los Angeles, with the Kings leading 4-0. They have never swept a five-game season set from the Oilers. . Edmonton is 6-24-3 when scoring fewer than three goals. All the wins have come in overtime or a shootout. . Hall has six goals in his last 32 games after getting the game-winner in consecutive games against the Sabres, Sharks and Rangers.