During a sometimes shaky opening period in their return from a four-day break, the Kings got a fortuitous goal when the 19-year-old Clifford deflected home Wayne Simmonds' shot in front of the net.
"It was an ugly goal, but it's good to get the first one off my back," said Clifford, who made the Kings' roster out of training camp but didn't score until his 22nd game.
The Kings' second-round draft pick last summer didn't even try to hide his broad grin while hugging his teammates and watching replays of his feat on Staples Center's massive overhead scoreboard. Clifford's early excitement gave a boost to his teammates through the rest of a grinding win over an opponent that has thrashed them regularly in recent years.
And when asked what he'll do with the puck, Clifford said: "I'm going to send it back home to my parents. Cheap Christmas present."
Anze Kopitar also scored a power-play goal for the Kings, who have won three straight after losing seven of eight in a skid that sent them tumbling from the top of the Western Conference standings.
"They've got some of the real tough guys in the league," Los Angeles coach Terry Murray said of the Flames. "Whenever you're playing against top goaltenders, you've got to get to the net. If you get pucks there, good things will happen."
Not everything was good for the Kings: Olli Jokinen broke up Quick's shutout bid on a power-play goal with 3:21 to play, also ending Los Angeles' streak of 45 straight penalty-kills at home from the start of the season.
Miikka Kiprusoff stopped 30 shots for the struggling Flames, who have alternated wins and losses in the past nine games. Calgary hasn't won two straight games since late October.
"I thought our defensemen were under siege a lot, and I didn't think our forwards had a good game," Calgary coach Brent Sutter said. "We weren't strong on our forecheck, and we allowed them to forecheck hard on us. We have to regroup and get going again. It sounds like a broken record, but that's the only way we can do it."
Calgary had won 14 of 17 meetings with the Kings, including a 3-1 home victory in October. Los Angeles won 11 of its next 13 games, moving to the top of the NHL standings before losing its grip.
The Flames thought they had scored a few minutes into the first period during a goal-mouth scramble. The officials' call on the ice was overturned after a lengthy video review in Toronto determined Matt Stajan had kicked in the puck with his right leg.
"I don't know if that's what I did," Stajan said. "It was a battle in the crease and I was trying to adjust myself because my stick was caught. Obviously when I turned, it looked like I kicked it. Trust me, I didn't know exactly where the puck was. There were guys falling into me."
After a scoreless second period, Kopitar connected early in the third when Jack Johnson's shot from the point deflected to him for the third goal in three games -- all game-winners -- for Los Angeles' high-scoring Slovenian forward.
Clifford was in the penalty box when the Flames finally scored. Anton Babchuk's shot from the point squirted through captain Jarome Iginla's screen and trickled behind Quick, where Jokinen slapped it home for his third goal.
After 12 straight home games of perfect penalty-killing, Los Angeles fell just short of becoming the first team to avoid allowing a power-play goal in 13 consecutive home games since Colorado did it in 2001-02.
"It wasn't everything, but we certainly were aware of it," Kings defenseman Rob Scuderi said. "We wanted to keep it going."
The game ended with a prolonged scuffle, and Calgary's Rene Bourque was sent to the penalty box with 5 seconds left for slashing. Kings defenseman Drew Doughty claimed it was worse, saying Bourque knocked the breath out of him by spearing him.
"You try not to be frustrated, but at the end of the day, it's always hard to swallow," said Iginla, who could be a trade target of the Kings if the Flames decide to break up their roster. "We have to keep going. This was another opportunity that got away, and we definitely feel it."
- LW Alexei Ponikarovsky returned to the Kings' lineup after a 12-game absence with a broken finger.
- Kings C Brad Richardson skated slowly to the bench in the first period with an unspecified upper body injury. He didn't return, and Murray didn't know what happened, saying Richardson will be re-evaluated Friday.