Sean Couturier scored, and Giroux had two assists for Philadelphia (24-23-7), which is eight points behind the Columbus Blue Jackets for the second wild card into the Stanley Cup Playoffs from the Eastern Conference. Stolarz made 37 saves.
It was the longest winning streak in the NHL and the longest for the Flyers since they won 10 straight from Nov. 27-Dec. 14, 2016.
"We didn't play our best game," Couturier said. "[Stolarz] was there to shut the door and gave us a chance to win. We got to do better in front of him."
Philadelphia thought it won the game in the third round of the shootout when the referee ruled Kempe's shot hit the crossbar. But video review showed the puck crossed the goal line.
"At the moment, I was let's say 90 percent sure it hit the white thing in the back (of the net)," Kempe said. "That was a good feeling obviously they confirmed it."
The Flyers were outshot 18-11 in the first period and fell behind 1-0 on Austin Wagner's goal off a give-and-go with Kyle Clifford at 8:46.
"I don't think anybody liked the first period," Philadelphia coach Scott Gordon said. "I thought the first half of it was OK. Then the last half we started playing slow, we didn't advance the puck, didn't get any puck pursuit in the offensive zone. Just real sloppy play."
Couturier tied it 1-1 at 7:00 of the second with his Flyers-leading 22nd goal. Giroux dove to whack a one-handed pass to Couturier in the slot, and he spun and backhanded a shot between Quick's skates.
Kempe finished a 2-on-1 with Brendan Leipsic to give Los Angeles a 2-1 lead at 12:33 of the second. It was Kempe's ninth of the season and fourth in a three-game goal streak.
The Kings are last in the Western Conference but are 5-2-0 in their past seven games to move within five points of the second Western wild card into the playoffs.
"We're getting closer and closer," Kempe said. "We haven't played good enough so far in the season, but I think the last couple weeks we've been starting to figure it out. The structure is there, the team game is really good there the past couple games. We've got to take that with us. Take it one game at a time."