Athanasiou put Chicago up 3-2 at 1:42 of the third period, tapping in a return pass from Lukas Reichel after Calgary forward Nazem Kadri turned the puck over in the neutral zone, and Wagner extended the lead to 4-2 at 3:11 when he toe-dragged around a sliding MacKenzie Weegar before shooting by Markstrom.
Backlund cut it to 4-3 at 13:09 when his centering pass into the slot deflected in off the skate of Blackhawks defenseman Ian Mitchell.
"We weren't playing with the pace, execution that we wanted," Backlund said. "I think urgency was not at the level you'd want us to be at in such an important game. Tied 2-2 going into the third, that's where we've got to push, and they did it to us instead. I thought the second we dominated the period, and we came back and had a chance to win the game in the third and gave up two right away."
Mrazek made a flurry of stops with under 30 seconds remaining, including an arm save on a slap shot from Toffoli with 29 seconds left to preserve the win.
"Eight-game losing streak, obviously not good, but I think it was a good response," Reichel said. "They're still in the playoff run, so we kind of wanted to give everything, give 100 percent. It was a good challenge for us. I'm glad we won."