Roslovic's goal came after Boone Jenner tied it 6-6 with 47 seconds remaining in regulation.
"Every time we'd score they'd come back and get one, but I liked how we just stuck with it and got rewarded tonight," Jenner said. "Gutsy win for us."
Nick Jensen scored twice, and T.J. Oshie and Sonny Milano each had a goal and an assist for the Capitals (33-31-8), who are 1-3-1 in their past five games. Charlie Lindgren made 32 saves.
Washington is five points behind the Florida Panthers, who have one game in hand, for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"Obviously, it's not coming easy right now," Capitals forward Tom Wilson said. "We shot ourselves in the foot a little bit tonight. Some of these games here can be kind of open hockey, and that's kind of what the other team wants, and we got to make sure we're keeping it tight and good habits, good fundamentals, and it just got away from us a little."
Adam Boqvist scored twice, and Emil Bemstrom and Eric Robinson each had a goal and an assist for the Blue Jackets (22-41-7), who trailed 3-0 in the first period. Daniil Tarasov made 31 saves.
"After the first period, I thought we played a solid game," said Columbus associate Pascal Vincent, who coached the team with Brad Larsen away due to a death in the family. "The guys kept fighting and fighting, and it was an interesting game. It was a fun game to coach. It was a fun game to be a part of."