The Kings rallied from a two-goal deficit in the third period on Tuesday, scoring three times to go up 4-3, but Washington rookie Evgeny Kuznetsov forced overtime with his first NHL goal inside the final minute of regulation.
An entertaining overtime period didn't produce a game-winning tally.
Kuznetsov had the puck roll off his stick to begin the shootout. Anze Kopitar then slipped a backhander through the pads of Capitals goaltender Jaroslav Halak.
The win was also the seventh in a row against Washington.
Dustin Brown and Dwight King each had a goal and one assist for the Kings, who remain eight points ahead of Phoenix for the third and final guaranteed playoff spot in the Pacific Division.
Los Angeles is one game shy of matching the longest road winning streak in club history from Dec. 18, 1974-Jan. 16, 1975.
Alex Ovechkin potted his NHL-leading 47th and 48th goals of the season for Washington, which earned at least a point in a sixth consecutive game (4-0-2) despite losing playmaking center Nicklas Backstrom to an upper-body injury in the second period.
"Obviously, you miss Backy, but every team has guys that go out," said Washington head coach Adam Oates. "You've got 19 other guys to do the job. We've got plenty of other guys that can do the job."
The Capitals are one of four teams tied with 80 points battling for the two wild-card spots in the Eastern Conference.
King scored 45 seconds into the third to trim LA's deficit to 3-2. Brown let a slap shot go from above the right circle and King pushed in the rebound.
The Kings tied things up when Marian Gaborik beat Halak high with a wrist shot from the bottom of the left circle at 8:41.
A giveaway by Washington resulted in the go-ahead goal for Los Angeles. Brown pounced on a loose puck in the slot and slipped it through Halak's five-hole at 12:55.
The Capitals managed to draw even while short-handed.
Halak was summoned to the bench once more and Kuznetsov tapped in a rebound of an Ovechkin shot that had gone between Quick's pads with 41.5 seconds remaining.
"We should have won it in regulation but just had a couple minor breakdowns on that goal," said Kings defenseman Drew Doughty.
The Capitals made the Kings pay for a pair of tripping penalties in the first period by converting on both power plays.
Backstrom was standing near the right post and Ovechkin sent a hard pass from the left half boards, hoping for a backdoor tap-in. However, the puck deflected in off Los Angeles defenseman Robyn Regehr at the 2:24 mark.
Ovechkin made it 2-0 when he wrapped up a passing sequence with a one-timer off Troy Brouwer's feed from the left circle at 7:04.
Los Angeles used its power-play unit to get on the board. Richards set up a screen in front of Halak, but was drilled by Alec Martinez's one-timer from the point. Richards gathered the puck, turned and swept it along the ice past Halak's right pad at 2:53 of the second period.
The Capitals had killed 22 straight power plays, but they regained a two-goal lead on former King Dustin Penner's first tally with Washington.
Martinez swept away Chris Brown's rebound bid, but wound up putting the puck right on Penner's stick and he deposited it into the open cage with 2:42 remaining in the second.