Atkinson scored the shootout winner and Mason — who didn't even start the game — made 28 saves as the Blue Jackets won 2-1 over Calgary.
"It's a nice win," said Mason. "It's not easy coming out to the West Coast, the travel, the time change, it all plays a factor into home ice advantage out here.
"To get a win here and get rewarded on the last leg of this trip is nice."
Starting goaltender Curtis Sanford, who had beaten Calgary twice this year, exited the game with a lower body injury with 5:23 left in the first period. He appeared to injure his leg while involved in a scramble to the side of the net.
The loss caps off a costly weekend for Calgary's pursuit of a playoff spot in the congested Western Conference. Riding a five-game winning streak entering the weekend, the Flames lost 3-1 to the 29th place Edmonton Oilers on Friday, before getting just one point against the last place Blue Jackets.
Calgary remains in 11th place in the West, one point behind San Jose and Los Angeles and two back of Phoenix and Colorado, who hold down the final two playoff spots.
"Our first was a little bit of a hangover from Edmonton where we weren't quite as sharp or as aggressive, or competed quite as shard but from the second period on, we played hard," said Flames captain Jarome Iginla. "We competed, we got pucks to the net, we hemmed them in and we need to keep that assertive style going for us. That's how we've got to play and it's the only way we're effective."
The Flames begin a crucial three-game road trip Tuesday night in Colorado.
"Obviously, it's a playoff game, it's a team ahead of us in the standings and as it stands right now, it's a game we have to look like we have to win," said Calgary defenceman Mark Giordano.
With neither team able to score in their first three attempts in the shootout, Atkinson, the Blue Jackets' 22-year-old rookie, scored by faking a shot, then going to his forehand to put the puck past Flames goaltender Miikka Kiprusoff.
"Obviously he's a talented player, skilled offensively, and he went in against one of the league's top goalies and really made a tremendous move," said Blue Jackets interim head coach Todd Richards.
Iginla was Calgary's last hope but just like Alex Tanguay, Olli Jokinen, and Curtis Glencross before him, his shot was stopped by Mason.
Defenceman Nikita Nikitin scored for Columbus (23-42-7), which was playing on back-to-back nights. The Blue Jackets lost 4-3 Saturday night in Vancouver.
Matt Stajan scored for Calgary (34-26-13), whose four-game home winning streak came to an end.
The Flames were being outshot 5-2 when Sandford was helped off the ice. As soon as Mason settled into the net, the home side turned on the pressure and carried the play most of the rest of the game including a 16-3 advantage on shots in the second period.
"We had lots of good scoring chances and we didn't give that many up. Fortunately, we didn't get it in the shootout and that stings a bit, but the guys played hard," said Iginla.
Calgary tied the game 1-1 at 8:34 of the third when Glencross centred a pass to Stajan, whose quick shot went off Mason's shoulder and in.
It was the seventh goal of the season for Stajan, four of them coming in his last six games.
"I don't think you can say we got outplayed tonight. We almost outshot them by double," said Stajan. "Sometimes we just have trouble finding the back of the net. It's not for lack of effort. We went hard the last two periods and overtime. It's just frustrating right now."
The only goal of the first two periods came from Nikitin at 13:03 of the first. Brett Lebda slid a pass from the blue-line to Nikitin, who one-timed a hard slapshot past Kiprusoff.
Calgary's roster had a different look with veteran Lee Stempniak back after missing 21 games with an ankle injury. However, Tanguay left the game immediately after getting elbowed in the head by Jared Boll in the Blue Jackets zone 6:31 into the second period. He did not return for the third.
The incident seemed to spark the Flames, who showed more emotion and greater urgency after that.
"In the second period, I don't think they got out of their end for more than more than two minutes," Iginla said. "Some days it doesn't go your way, but we keep our heads up, we get ready for Colorado and we play them and we can tie them and keep going."
Notes: Columbus wins the season series over the Flames 3-0-1. ... Calgary has lost its last four shootouts and falls to 3-8 on the season. Columbus improves to 4-5 in shootouts. ... Five of Nikitin's 23 points have come against Calgary (two goals, three assists) this season. He was the point leader for both teams in the four-game season series. ... Stajan played in career game No. 600... Flames see their record drop to 7-11-2 against the Central Division..