After an even first period, Kjellberg gave Anaheim a 2-1 lead 27 seconds into the second when he advanced in the middle and flicked the puck over goaltender Milan Hnilicka, who later was pulled.
Kevin Sawyer made it 3-1 more than six minutes later with his first goal of the season and second career.
"These are the kinds of games where you've got to make sure you come out ready to play and not take anything lightly," said Sawyer, who skated wide around the net and pushed it in off Hnilicka's side for his first goal since Nov. 25, 2000 against the New York Rangers.
With 3:30 left in the second period, Niclas Havelid made it 4-1 with his second goal this season. Steve Rucchin and Paul Kariya, Anaheim's top scorers, combined to set up the tally that chased Hnilicka in favor of Pasi Nurminen. Rucchin and Kariya each provided two helpers.
Hnilicka faced 20 shots in 36:30 minutes played.
Peter Sykora added a third-period goal for Anaheim, which bounced back nicely from a tough 2-1 overtime loss at Detroit on Friday.
Sykora's shot was the only Nurminen conceded of 13 faced. On Friday, Nurminen was replaced by Hnilicka in an identical loss
Vyacheslav Kozlov gave the Thrashers the lead on the power play at 4:56 of the first period with his fourth goal of the season.
Anaheim goaltender Jean-Sebastien Giguere stopped 27 shots to even his record at 7-7-2.
Celebrity Families of 2014 [PHOTOS]