Pittsburgh has won its past three games by a combined 14-3 margin.
Fleury, who has 30 career shutouts, made 16 of his 36 saves during the third period to help preserve a two-goal Pittsburgh lead established in the first period. Chris Kunitz scored an insurance goal in the third, his second of the game, when he collected a rebound off of a shot from Sidney Crosby and wristed the puck from a sharp angle past Kings goalie Martin Jones at 9:24.
"I think we faced a good team," Fleury said. "Even though they're missing a few key forwards, they're still the Stanley Cup champs. So, it was a good challenge for us, and we played a solid game."
The Kings gave credit to Fleury, but several Los Angeles players said they thought they failed to make the Pittsburgh goalie work enough.
"[Fleury] made the saves he was supposed to make," Kings forward Dustin Brown said. "I don't think he really had very many hard saves, quite honestly. He got to see the puck. I think our best chances came probably late in the game on the power play, when we had some guys around the net and rebounds, but we didn't make it hard enough for him."
The game matched the physical and defensive style of the Kings, with Los Angeles outshooting Pittsburgh 36-23, but the Penguins were able to withstand the Kings' grit while showing some of their own.
With five seconds remaining on a 5-on-3 with Tyler Toffoli and Drew Doughty in the penalty box, Evgeni Malkin found Kunitz charging toward the net. Kunitz tipped Malkin's slap pass by Jones 9:16 into the first for a 1-0 lead. It was Malkin's 400th career assist.
Malkin, who has scored at least one point in each of Pittsburgh's nine games, is the fifth player in Penguins history to reach 400 assists, joining Mario Lemieux (1,033), Jaromir Jagr (640), Ron Francis (449) and Crosby (503).
"I'm just doing my job right now," Malkin said. "I work with a great line with [Pascal Dupuis] and [Blake Comeau], and we try to use each other. It's a good power play too, most points [came] on the power play, but a few chances 5-on-5 too.
"It was a good game, and I like to help the team win."
Kunitz helped generate the Penguins' second power-play goal of the first period when he sent a pass through the Kings crease to Kris Letang in the right faceoff circle. Letang's shot deflected off of Los Angeles forward Dwight King's stick and through Jones' legs to extend Pittsburgh's lead to two goals with 4:46 left in the opening period.
Kings coach Darryl Sutter said he thought penalties cost Los Angeles the game.
"I know there's a big deal made about the offensive juggernaut that they are," Sutter said. "But there's only a couple goals difference, 5-on-5 coming into the game."
The Penguins have converted an NHL-best 39.5 percent of their power-play chances, 13 percentage points clear of the second-ranked Arizona Coyotes.
"We knew with their power play, it's obvious what the percentages are and how many goals they've scored on the power play, compared to 5-on-5," Kings center Jarret Stoll said. "We stay out of the box, we play 5-on-5 with these guys, we play stingy, we check well, we'll get our chances. That's how we needed to play this game, but we didn't.
"Obviously, we lost and we should've lost."
The Kings controlled the second period, holding the Penguins to six shots despite Pittsburgh having three power plays. But Los Angeles couldn't take advantage of its 10 shots in the period.
Pittsburgh managed one shot on its third power play of the second, and the Penguins allowed several shorthanded chances during their second chance. Most of the period was spent in the Penguins zone, but the Kings were primarily kept to the outside, with the Pittsburgh defensemen guarding the crease.
Fleury made 10 saves in the second, but few of the shots were real threats to get past him.
The Penguins, who had the League's lowest-ranked penalty kill through the first few games of the season, killed each of the Kings' two power plays through the first two periods and two more in the third. They have killed 22 consecutive penalties and are ranked 11th in the League with an 83.8 percent success rate.
Penguins coach Mike Johnston said Pittsburgh tried to force Los Angeles into penalties and he felt his players successfully pressed the Kings into mistakes.
"I think in games, you try to earn penalties," Johnston said. "I've found that in some of the games lately, we haven't earned any penalties. We haven't fought for space. We haven't drove to the net hard, and when you do those types of things, you tend to draw some penalties, and I thought we earned some penalties tonight.
"They probably didn't want to give us that many, with our power play. But a lot of the times, you fight for space and a guy has to make a choice. He either has to let you go or else he's going to have to interfere with you."