Alexander Radulov scored the only goal in the shootout in the forward's return after missing 15 games with a lower-body injury.
Joe Pavelski scored for Dallas (9-9-6), which is 3-1-2 in the past six games.
It was the first win after regulation for the Stars in the past seven tries. They are 2-6, including 1-3 in the shootout.
"We know we've had some bad breaks in overtime this year," Oettinger said. "It's just the way it goes, but to get the one tonight is huge, feels great, and just a great road win."
Zach Werenski scored, and Joonas Korpisalo made 34 saves for Columbus (11-12-7), which defeated Dallas 4-3 in overtime Saturday.
"We created plenty of offense, but in the end, it just wasn't enough," Werenski said. "We're definitely hitting our stride. This weekend was a great step forward for the rest of the year."
Oettinger is 4-1-5 in 13 games this season. He is 3-0-1 against the Blue Jackets, including a 5-0 win March 6 when he made 21 saves for his first NHL shutout.
"He made some big saves," Pavelski said. "Just solid, kind of like he's been all year."