The home team Dallas started goal scoring in the first period. At the start of the second, Vegas could begin to turn, when Karlsson was served by Jonathan Marchessault a puck as the Swede thumped behind Ben Bishop in the Dallas goal.
It was the 26-year-old's tenth goal for the season.
The team then had to score each other for the regular time, and the match was decided by Vegas Max Pacioretty 51 seconds into the extension.
It was the Texas team's second game under temporary coach Rick Bowness, after Jim Montgomery was fired Tuesday.
The loss means a victory suite is broken for Dallas after three games. Montgomery had to go because of unspecified "unprofessional behavior", not because the team played poorly