Steve Ott scored at 2:56 of overtime, Loui Eriksson notched the tying goal in the final minute of regulation, and the Stars rallied for a 3-2 victory over the Kings on Wednesday night.
Quick had won his previous six starts against Dallas and the Kings were 8-1-1 in their past 10 road meetings with the Stars.
The Stars had dropped their first two matchups with the Kings this season, and needed some late-game scoring to end the domination.
"We stuck with it," said Ott after his first career regular-season overtime goal and second straight game-winner.
"I thought the game plan coming in was pretty well taken care of throughout the whole 60 minutes. The unfortunate part was we were down 2-0 so we had to come back out of a little bit of a hole. I think we out-chanced them a lot tonight and I think the final result was probably the proper result."
With the game tied at 2, Stephane Robidas skated into the Kings zone and flipped a centering pass to Ott, who went hard to the net up the left side and fired a shot past Quick.
Eriksson tied it with 20.3 seconds left in regulation with his 10th goal, capitalizing on a set play drawn up by associate coach Willie Desjardins after a timeout.
Eriksson took a cross-ice feed from Jamie Benn and beat a sprawling Quick.
"Willie drew up the play there," Stars coach Glen Gulutzan said. "We talked about just going forward on the draw and jumping them as quick as you could and when you've got guys like [Benn] out there and Loui that can make plays. They read off it pretty good."
The Kings had a bad break on the play when the puck bounced off Los Angeles defenseman Willie Mitchell, giving the Stars possession deep in the offensive zone.
"They had three guys lined up on the inside and they go forward," Kings coach Terry Murray said. "It hits Mitchell right in the midsection. The puck just bounced down and they took a whack at it, went to their guy on the outside and he makes a play. It was nothing more than that."
The Stars went on the power play with 2:21 remaining in regulation when Ethan Moreau was penalized for slashing.
Goaltender Kari Lehtonen was pulled for a sixth attacker, but the Kings killed off the power play and appeared to be set up for another win over the Stars.
"It's frustrating," Kings center Anze Kopitar said. "You got a two-goal lead going into the third period and you couldn't close it out. They kept coming. You got to give them credit, they made some plays, but on our part, we just can't sit back like that. We got to play our style and keep the foot on the gas pedal and not back out."
Benn added a goal for the Stars, who've won two straight after losing five in a row.
Mike Richards stretched his goal-scoring streak to five games and Quick stopped 35 shots for the Kings.
Richards and Kevin Westgarth connected during a 36-second span of the second period.
Richards put the Kings in front on a shot from the slot that dribbled through Lehtonen's pads and over the goal line at 4:53, Richards' 10th of the season.
Richards has eight goals in his past eight games.
Westgarth registered his first of the season at 5:29 of the second when his shot from the left circle whistled past Lehtonen's left shoulder and into the upper right corner of the net.
Benn pulled the Stars to 2-1 at 2:07 of the third period on a shot from low in the left circle, Benn's sixth of the season.
Los Angeles is without LW Dustin Penner (quad), D Alex Martinez (upper body) and LW Scott Parse (lower body). ...Stars LW and captain Brenden Morrow was scratched (upper body injury) and is day to day. ... The Stars are 12-0 this season when allowing two goals or fewer.