Matthew Tkachuk sat out his third straight tilt, listed as unable to play.
The Flames, of course, were coming off a tough 5-4 overtime defeat in Game 4 on Sunday afternoon.
The Flames powerplay got an early crack at the Stars, when big Jamie Oleksiak was tagged for a blatant hold on Johnny Gaudreau, wrapping the Flames star in a hug and spinning him down.
However, Calgary wasn't able to manage a shot on the advantage.
The speedy Denis Gurianov - who scored in Game 4 - later broke down the wing and managed to stay ahead of a diving Erik Gustafsson, cutting towards the net but losing the handle on the puck.
As he circled back out front, Rasmus Andersson was called for hooking. That advantage lasted just 41 seconds before Alexander Radulov was sent off for interference on Mikael Backlund as Dallas tried to gain the Calgary zone.
The Andersson penalty had just expired - to the second - when Stars captain Jamie Benn opened the scoring, converting a pass from Tyler Seguin on the rush - and then barrelling into the net with Dillon Dube backchecking hard on him.
It was the 10th shot of the game for the Stars, who scored first for the second straight outing. The Flames, at that point, had managed two shots on Anton Khudobin, who started again for big Ben Bishop, who was listed as unable to play.
Mark Giordano came close to tying it up with a laser on a PP after the Stars were busted for Too Many Men, but wired it off the post.
Backlund showed some poise and patience to even things up with less than a minute left in the frame, pausing with the puck as he skated into the high slot - letting Esa Lindell swing past him attempting to block his shot - before snapping it through traffic glove-side past a screened Khudobin, who had both Tobias Rieder and Andrew Mangiapane parked on his doorstep.