Trevor Lewis and Tanner Pearson scored for Los Angeles (12-7-2). Jonathan Quick was pulled in the first period after allowing the three goals on nine shots and lost his fifth straight start. Darcy Kuemper replaced Quick and finished with 30 saves.
The Kings have lost five of their past six games.
"They came after us right away and obviously got the three-goal lead," Lewis said. "We battled back, but I think we kind of [hurt] ourselves in the first period."
Karlsson, who has scored all nine of his goals in the past 13 games, scored 55 seconds into the first period from the left circle to make it 1-0. His second goal at 11:22 in the first period led to Quick's exit and gave Vegas a 3-0 lead.
"Those are pucks [Quick] would love to have back," Kings coach John Stevens said. "It's a team game. Johnny's bailed us out many of nights. That's certainly not the start we wanted. You dig a hole on the road against any team, especially in here, it's tough to come back."
Karlsson said he was shocked to see the puck go between Quick's legs as easy as it did for his first goal.
"I'll take that any day of the week," he said.
Reilly Smith assisted on each of Karlsson's goals.
Eakin scored his third goal of the season at 9:21 of the first period.
Lewis made it 3-1 at 8:35 of the second period. Erik Haula lost control of the puck at center ice, and Lewis deked around Lagace for his sixth goal of the season.
Pearson scored at 11:45 of the third period to make it 3-2, but Tuch scored an empty-net goal with 1:02 remaining.
