Mark Letestu snapped a 1-1 tie in the third period for the game-winner.
Before the game ended, the Stars were eliminated from playoff contention when Detroit defeated Nashville 5-2, moving past Minnesota into seventh in the Western Conference.
"It hasn't really sunk in yet, I don't think," said goaltender Kari Lehtonen, who made 32 saves in a losing effort. "It feels bad right now and I'm sure it feels even worse tomorrow."
The Blue Jackets are even in points with the Wild, who hold eighth place due to the non-shootout wins tiebreaker.
Columbus needs to beat the Predators on Saturday, and have the Stars beat the Red Wings in regulation on Saturday, or have the Wild earn no more than one point in their two remaining games, to earn just the second postseason berth in franchise history and first since 2009.
"Our players have earned this opportunity," Blue Jackets coach Todd Richards said. "Granted, we still need some help, but it's going to make for a great environment on Saturday in our building."
Making the playoffs would cap off a remarkable turnaround for Columbus, which started the season 5-12-2 and was last overall in the NHL in late February, before reeling off an 18-5-5 run since to make an improbable bid for a postseason berth.
"It feels great. You can only control what you can control and that's what we're doing," said Atkinson, who had a game-high seven shots on goal. "We're doing our job and that's all we can do. I felt like I probably should have had six or seven tonight but I scored when it counted. It's exciting to score but it's even better to win games."
The Blue Jackets, which concluded a six-game road trip with a 5-1-0 record, got two third-period goals from its league-worst power play to secure the triumph.
"We came in here, did our job and we won our game," said center Brandon Dubinsky, who added two assists. "We're going to have to do it again on Saturday. Hell of a job by these guys, six-game road trip, 5-1 against some damn good teams, really proud of these guys. The job's not done, but we gave ourselves an opportunity."
Jamie Benn scored for the Stars, who have lost four straight (0-3-1) to miss the playoffs for the fifth straight season, the longest drought in franchise history.
"We were in every single game and the guys battled so hard for that last stretch," winger Eric Nystrom said. "We just came up a little short and got zero help -- it was just the combination of everything going against us. We got so many good, young guys coming up, so much character, that we're going in the right direction."
The Stars knew their fate by the time the third period started and that's when Columbus seized control of the game.
"To be honest, we knew after the second period and I think most of our players did too," Dallas coach Glen Gulutzan said. "However, at the moment, the loss stings. You're focused on this game. And we were a little flat in the third, maybe for a reason though."
Letestu's 13th goal of the season snapped a 1-1 tie at 7:56 of the third, just three seconds after Stephane Robidas entered the penalty box for an interference penalty. Dubinsky won the faceoff, pulling the puck back to Letestu at the right point for a slap shot that appeared to nick off Nystrom's stick and sailed over Lehtonen's glove.
"If they win the draw, it's 30 seconds off the kill. So he makes a good draw and I just try to get it on there as quick as possible," Letestu said of Dubinsky. "We try to make an effort to get a lot of pucks on the net on the power play and it just happened to hit Nystrom's stick on the way through and changed direction."
The goal snapped a 1-for-22 power play drought for Columbus.
Bobrovsky came up with a big pad save at 10:10 to preserve the Columbus advantage, denying Ray Whitney's wrist shot from the slot.
A Nystrom interference penalty with 2:49 remaining put the Blue Jackets back on the power play and Atkinson capitalized on the opportunity with 1:40 to go.
After Lehtonen stopped Jack Johnson's shot from the left point, the rebound came out to the left of the crease, right to Atkinson, who tucked it just inside the left post for his eighth goal of the season.
"It ended up being the difference in the game," Richards said of the power play. "We got two and we had changed some things. And we had a good play there in the third period, shooting the puck, simple plays, getting the puck to the net."
After falling behind on the game's opening shift, the Stars tied it on the power play at 8:43 of the second period. Erik Cole's pass into the slot from the right faceoff circle deflected off Atkinson's stick and bounced right to Benn at the left post for an easy tap-in.
Benn's 12th goal of the season -- and fourth in the past six games -- swung the momentum Dallas' way and it continued to press throughout the period, but Bobrovsky shut the door.
The Blue Jackets jumped on top immediately, as Atkinson scored just 19 seconds into the opening period. Lehtonen stopped Blake Comeau's wrist shot from the slot, but left a rebound next to him in the crease and Atkinson pounced on it, jamming it home for a 1-0 lead.
"Of course, that's not what you want to do right on the first shift," Lehtonen said. "I felt like I made a couple of good saves right before that, and then I just lost the puck."
Bobrovsky, considered a Vezina Trophy candidate as the NHL's top goaltender, entered with a 16-5-3 record since March 1, with a 1.65 goals-against average and a .945 save percentage, as well as four shutouts. ... The Stars honored former D Richard Matvichuk as part of their 20th anniversary all-time team. Matvichuk, who played in Dallas from 1993-2004 as part of a 14-year NHL career, was part of the franchise's only Stanley Cup team in 1999.
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