Handzus hammered a shot from the left side of the slot past goalie Alex Auld. to give the Kings their second win in a row over the Stars.
Kopitar completed his hat trick in the first two periods, helping the Kings build a 4-1 lead before Dallas rallied with three goals in the third period.
James Neal tied it at 2:46 with his second goal of the game.
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 Kings goalie Jonathan Quick off guard. Then, Mark Fistric fired a shot from the blue line that went between Quick's legs.
Wayne Simmonds also scored, and Alexander Frolov had two assists for Los Angeles. 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.
The Kings are 5-1 this season and 44-22-8 overall when Kopitar, the first Slovenian to play 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.
Kopitar added two more in the second period, scoring on a power play and then on a nifty backhander that Auld couldn't control.
Frolov proved a valuable part of the offence after sitting out Monday night and being criticized by coach Terry 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.