San Jose's Torrey Mitchell opened the scoring 2:11 into the first period, and new Sharks captain Joe Thornton made it 2-0 on a power play midway through the opening period.
"It was kind of a lucky one. I think it tipped off a defensman's stick," Thornton said. "For the guys to bear down and get the two points tonight was very important."
The teams will meet again in Stockholm on Saturday night.
The Blue Jackets had three Swedes in their lineup, and the Sharks two.
"It's good to be home, I was a little nervous today," said Niclas Wallin, San Jose's Swedish defenseman. "I had a lot of people down here."
It's the third straight year that NHL teams have played in the Swedish capital, and there were a mixture of jerseys from teams across the league among the near-sellout crowd of 13,500 at the Globen Arena.
Much of the action happened in the frantic first period. And the intensity of the regular season traveled across the ocean, with a couple of fights that had the fans on the edges of their seats yelling and cheering.
The Sharks topped the Western Conference with 113 points last season, but their playoff run end in the conference finals when they were swept by Chicago.
It was Columbus coach Scott Arniel's first game in charge of the team.
"We did lots of good things," Arniel said. "Five on five I really liked the way we played. We gave up on goals-scoring chance in the third period and they scored their goal."
Former Vancouver captain and Swedish star Markus Naslund received a standing ovation and the loudest cheers of the evening when he dropped the puck during a ceremonial faceoff.