After the Stars overcame a 4-1 deficit with three third-period goals, Handzus hammered a shot from the left side of the slot past goalie Alex Auld to give the Kings their second successive victory against the Stars.
"I stress with every team, if you play with a checking mentality, good things happen," Kings coach Terry Murray said. "You end up with the puck on your stick at a critical point and it happened that way for [Handzus] tonight."
Dallas' James Neal tied it at 2:46 to go with his second goal of the game.
"You never want to put yourself in the hole like we did, but it shows the character that we have," Neal said. "When you come back like that, any little bounce in overtime is going to win it. Tough loss, but it's a good character comeback for us."
Dallas scored twice in under two minutes to cut the Kings' lead to a goal. Jamie Benn scored on a backhander at 6:58, catching goalie Jonathan Quick off guard. Then, Mark Fistric fired a shot from the blue line that went between Quick's legs.
"It was a good game, right, then the good old Kings came out in the third with a three-goal lead and starting sitting back, waiting for them to come," Kopitar said. "We cant play like that."
The Kings, tied with Phoenix for the Pacific Division lead, have won three of four home games this season. On Monday night in Dallas, the Kings beat the Stars 4-2 to snap a three-game losing streak.
Los Angeles is 5-1 this season and 44-22-8 overall when Kopitar -- the first Slovenian to player in the NHL -- scores. He leads the Kings with eight goals this season.
Kopitar opened the scoring with a highlight-reel goal. Receiving the puck on the right side, he shook off defender Stephane Robidas and skated across the front of the crease with the puck before slipping it past Auld on the far left side.
"That's a hockey-player play," Murray said. "That's a powerful play, and it takes a very courageous player to make that kind of strong move. Boy, for a young guy to start playing that and showing that kind of performance, that's pretty exciting stuff."
Kopitar added two more in the second period, scoring on a power play and then on a nifty backhander Auld couldn't control.
"His scoring is what's expected, but it's everything else he's doing that is taking him to the next level," Kings winger Dustin Brown said. "He's a guy we've needed and we can't ask for much more."
Frolov proved a valuable part of the offense after sitting out Monday night and being criticized by Murray for a lack of effort.
Frolov responded well, making an impact on the ice. He started quickly with an assist on Simmonds' goal at 15:37 of the opening period. Frolov broke free down the left side and found Simmonds, who backhanded the puck past Auld to make it 2-0. Frolov also assisted on Kopitar's third goal, which ricocheted off Auld's glove.
"He played really well," Brown said about Frolov. "He's a pro guy."