Jonathan Toews scored the winning goal off his own skate at 14:33 of the third period, giving Chicago a 4-3 victory on Monday night to end a three-game skid.
"Necessary win, huge win for us," Chicago coach Joel Quenneville said.
The defending Stanley Cup champions swept the Kings in four games for the first time since 1981-82, when they went 3-0-0 against them.
Patrick Sharp continued his success against Los Angeles, scoring two goals and assisting on another. His three points gave him eight this season against the Kings.
"Every time we play those guys, it's a tough game and it goes back and forth, like it was tonight," he said. "I know we've gotten the better of them a few times, but they're a team that's certainly not going to go away. They're going to be competitive all year long."
Sharp's two goals gave Chicago the lead both times until Ryan Smyth tied it 3-3 for the Kings on a drive that broke his stick at 12:42 of the third.
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"We stuck with it. It wasn't the prettiest game on our part after 40 minutes," Sharp said. "But our power play kept us in it along with our goaltender, and we kind of regrouped and raised our battle level. It was nice to see us go out there and score a couple of 5-on-5 goals."
Toews answered 1:51 after Smyth's goal, when he scored in traffic after Jonathan Quick had already made one save, drawing cheers from a contingent of red-clad Hawks fans sitting behind their bench.
"That's bad luck," Los Angeles coach Terry Murray said. "It almost went off Quick's head when it came off the crossbar. I guess when you're the Stanley Cup champions, you get lucky breaks like that. We shouldn't have put ourselves in that position."
Corey Crawford made 26 saves as Chicago avoided a four-game skid for the first time this season.
"We knew it was going to be a tight game, so we just stuck it out. But we want to be better defensively than we were," Toews said. "There were some lapses, especially on the penalty kill, that weren't good for us. But we just found a way and that's the main thing."
Toews returned after missing two games because of a right shoulder injury.
"He brings a lot to the table. One of the biggest things is his grittiness and his ability to play in front of the net low and know where everyone is going to be," teammate Duncan Keith said. "You can never have enough of that, and sure enough, he gets the game-winner being right in the paint and making a great play."
The Kings dropped their fourth in a row, giving them their second such losing streak this season.
"We're definitely close [to the Blackhawks], but close isn't good enough. We have to find a way to win," Anze Kopitar said. "We can pull together a string of shifts where we can dominate any team in the league. But for some reason, we let up. Sometimes we try to do too much and don't do the right things, and that burns us every time."
Sharp's second goal made it 3-2 early in the third off Alec Martinez's turnover. Smyth tied it on a power play after Toews was sent off for high sticking.
Sharp's first goal, a power-play score, gave the Hawks a 2-1 lead at 7:11 of the second. Smyth tied it on a breakaway at 18:17 of the period.
Chicago tied the game at 1-all on Marian Hossa's power-play goal at 18:42 of the first. Hossa drove to the net with none of the Kings stopping him for his ninth goal of the season and first since Nov. 14 against Anaheim.
Handzus' goal was his first since Nov. 20 at Boston. ... The Hawks are 6-2-0 in back-to-back games, their eighth of 19 such sets this season. Only Minnesota will play more back-to-backs (19) in the Western Conference. ... Sharp has 13 multipoint games, a team high.