Nick Bonino also scored and Adin Hill made 24 saves for the Sharks, who have won three straight and five of six. Timo Meier had two assists for his fourth multipoint game in the last five, giving him five goals and three assists during that stretch.
San Jose, in the midst of its third five-game road trip this season, improved to 8-5-0 away from home.
''I like our road mentality and perseverance,'' Sharks coach Bob Boughner said. ''We have to grind teams down to win. We don't have Connor McDavid and Leon Draisaitl. We have to manage the puck and manage leads.''
Mathew Barzal scored early for the Islanders and Ilya Sorokin stopped 33 shots. New York, which had lost eight straight in regulation, is winless in five games at UBS Arena, its $1.1 billion new home next to Belmont Park.
The Islanders were playing for the first time since last Friday after having two games postponed due to the team's COVID-19 outbreak.
''It's a long time since we got a point, so we'll take the point,'' New York coach Barry Trotz said. ''It's a step in the right direction. ... We've just got to worry about the process of playing better and better.''
In the extra period, Karlsson skated up on a 2-on-1 before passing to Meier, who cut from left to right across the front of the net but couldn't get a shot off. Meier then passed the puck back from the right side across the front and Karlsson put it past Sorokin for his fifth of the season.
''I was screaming at (referee Dean Morton) because he didn't call a penalty, but I kept my eyes on Timo and he made a perfect pass,'' Karlsson said. ''This time of year is a lot of grinding, lots of traveling, and everyone is helping each other out. ... Today was a perfect example of that.''
Sorokin came forward to stop a shot by Alexander Barbanov 7:47 into the third period, and Hill gloved a shot by Barzal on a breakaway 10 seconds later to keep the score tied.
''Not a lot of their shots that came from the blue line got to the net,'' Hill said. ''We like playing those tight, defensive games. Our guys did a great job.''
The Sharks outshot the Islanders 14-7 and had ample chances in a scoreless and fast-moving second period.
Sorokin had to make a flurry of saves in the opening minutes of the period. With the goalie down on the ice, Rudolfs Balcers tried to lift the puck over him, but it hit off Islanders defenseman Noah Dobson to keep the score tied.
Sorokin stopped a slap shot by Karlsson and denied rookie Jonathan Dahlen midway though the middle period. He also gloved a high shot by Meier with 4:47 left in the period.
Barzal got the Islanders on the scoreboard 1:04 into the game on the first shot for either team. He brought the puck up the right side, skated into the right circle and beat Hill up high on the blocker side for his fifth. It gave the Islanders their first lead in five games in their new arena.
''We had a really good start,'' Trotz said. ''We created a lot, put pucks in play, we managed the puck well. Then we gave up chances in flurries. ... Our goaltender was outstanding during those flurries.''
Bonino tied it at 9:10 as he was standing in front of the net and redirected a pass from Dahlen from the right side with Sorokin out of position for his third.
''I think I said `thank you' to everyone involved in that goal about three or four times,'' Bonino said. ''Getting that goal gives you some confidence and allows you to focus on your game. ... My game is to keep it simple. The Islanders are obviously struggling. They score on the first shot and Hill was amazing after that.''
Anders Lee nearly gave New York the lead again, but his attempt in the final minute of the first hit off a goalpost.
The Islanders got several players back from their COVID-19-related absences. Josh Bailey returned after missing five games, Lee and defensemen Adam Pelech and Andy Greene each missed four. Kieffer Bellows took part in practice but sat out, and defenseman Zdeno Chara skated on his own.
Sharks forward Jonah Gadjovich left the game 3:11 into the first period after taking a shot by Dobson off his leg. Gadjovich was down for several minutes before being helped up and off the ice.
Sharks: At the New York Rangers on Friday night.
Islanders: At the Detroit Red Wings on Saturday night.
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