Connauton scored 3:57 into the second period to make it 3-0, beating Stars goaltender Anders Lindback with a slap shot from near the blue line that struck the right post before crossing the goal line.
Connauton played 44 games for Dallas, including eight this season, before the Stars placed him on waivers and the Blue Jackets claimed him Nov. 18. All six of his goals have come in 18 games with Columbus.
"It's awesome," Connauton said of the win. "This is a fun time right now. We're stringing together quite a few wins in the last month or so, making a good push. Everyone in here, we've got the same goal: We want to make the postseason. We're just taking it step by step, and that was a big win for us tonight."
Nick Foligno had three assists, Ryan Johansen a goal and an assist, and Scott Hartnell two assists for Columbus (18-17-3), which got 24 saves from Sergei Bobrovsky. Mark Letestu scored late in the third period to clinch the fourth straight win for the Blue Jackets in Dallas.
"You can see what [Letestu] adds to our team, and same thing with Cam (Atkinson) back into the lineup, his speed and his energy off the rush. Both guys make us a faster team," Columbus coach Todd Richards said.
James Wisniewski also scored for the Blue Jackets, who are fifth in the Metropolitan Division with 39 points.
"It's only about winning games right now," Richards said. "It's not about what are we, where are we? We'll enjoy it tonight, enjoy it tomorrow, fly up to Toronto, have a good practice on Thursday and get ready for a big weekend."
"I thought we got off to a real poor start," Stars coach Lindy Ruff said. "We weren't ready to play. Taking three penalties in the first, that hurt us. After that, we were the team that dominated possession."
Columbus took the lead 5:21 into the first period when Wisniewski scored his fourth goal on a shot that deflected off Hemsky's stick and went through Lindback's five-hole.
Lindback was making his fifth start of the season and first since Nov. 29.
"It's tough. He's had a long spell (between games)," Ruff said. "We probably needed another save out of him. He committed a little too early on a couple, and it's not easy."
A key for Columbus was keeping the potent Dallas power play off the scoresheet; the Stars were 0-for-5.
"Three (games) in four (nights), and you're relying on your PK to do the job, but we didn't complain. We got the job done," Foligno said. "Our D were outstanding, and then our forwards did a great job of killing those off. That's a huge part of the game, and we were able to do it. That's the reason we got the win."
Ruff wasn't happy the Stars were blanked in five power-play chances.
"The power play had spurts, but not good enough," Ruff said. "Could have been the difference-maker, and just couldn't quite get it going. Too many turnovers, too many giveaways, too many blind plays.
"Both teams' penalty killing -- we made a mistake on an entry, but other than that I thought our penalty killing was really strong. We had maybe three opportunities to score on our power play, which with that number on power plays isn't enough."
"I thought Ritchie played real well," Ruff said. "He might have been our best player tonight. I don't know if he drew three, four penalties. (He) skated, physically dominated. I thought he had a lot good going for him."
The Blue Jackets took a 2-0 lead at 7:26 of the first period when Johansen beat Lindback top shelf from the edge of the right circle with Stars forward Travis Moen in the penalty box for interference. Hartnell delivered a solid cross-ice pass that Johansen quickly converted for his 13th goal.
Johansen has a point in eight consecutive games.
Dallas answered with goals from Hemsky and Roussel to trail 3-2 at the second intermission.
Hemsky scored his fifth at 5:28 with a wrist shot from the right circle to beat Bobrovsky, who was screened by the Stars' Jason Spezza.
Roussel scored his 10th goal with 6:46 remaining in the second period when a wrist shot by Ryan Garbutt from the slot deflected in off Roussel's skate. Garbutt intercepted an ill-advised backhand pass by Atkinson inside the right circle before sending the puck toward the net.
"I was at the net. Lucky bounce. I'll take those any day," Roussel said.
Columbus sealed the win with 2:58 remaining in the third period when Letestu, playing his first game since Oct. 28 because of a groin/abdominal injury, capped a give-and-go with Alexander Wennberg with his fourth goal.
"As far as I'm concerned, [Wennberg] is the most skilled player for that 2-on-1," Letestu said. "I was rewarded by somebody's incredible talent."
Lindback made 16 saves.
"I think I played pretty well," he said. "Obviously, when you let in four goals, you can't be too happy about it and you lose a game. Obviously, I'm disappointed about it."