It was Hayes' first goal of the season in his second game; he made his debut in a 5-2 loss at the Dallas Stars on Saturday after missing 12 games recovering from Sept. 21 abdominal surgery.
Hayes pointed to the sky after scoring as a tribute to his older brother, former NHL forward Jimmy Hayes, who died Aug. 23 at age 31.
"It was a big goal," Hayes said. "The trainers got the puck for me. I'm going to give it to Jimmy's kid, Jimmy's son, Beau. It'll be one I'll remember forever."
Atkinson, who was Jimmy Hayes' roommate at Boston College, was on the ice for the goal.
"I think everything happens for a reason, and I was the first guy to hug him," Atkinson said. "Got the puck for Kevin. It's a big goal not only for our team, obviously, but for Kevin. I think we'll all remember this for a long, long time. ... I know how special this goal was for Kevin. I was just happy to be part of it."
Oliver Kylington put the Flames ahead 1-0 at 10:31 of the first period when his point shot went in off Couturier.