Islanders goalie Kevin Poulin lasted just 1:35 into the game before he got the hook, giving up goals to Karlsson and Jason Spezza 27 seconds apart.
Craig Anderson backstopped the Senators' win, turning aside all 28 shots he faced for his third shutout this season.
Al Montoya gave up four goals on 27 shots in relief of Poulin.
The Islanders have alternated losses and wins in their last nine games.
Celebrity Families of 2014 [PHOTOS]