Backlund and Troy Stecher scored, and Markstrom made 27 saves for the Flames (37-27-17), who trail Winnipeg by two points with one game remaining but can't win the tiebreaker because of fewer regulation wins (35-30).
"It's still so fresh," Calgary defenseman Rasmus Andersson said. "I think coming home tonight and waking up tomorrow you're going to realize what an opportunity we missed. Now we can hardly believe it. It's painful. I think coming home tonight and waking up tomorrow, that's when you're going to realize we're out and it [stinks]."
The Flames are 18-13-17 in one-goal games this season.
"This year, looking around this room, we have the squad to do it and that probably makes it more frustrating, just we didn't do it," Andersson said. "Being in so many games and not coming out on the winning side of those, that's probably the one thing that's the most frustrating, for sure."
Sanford gave Nashville a 2-1 lead at 4:02 of the third period when he banged in a loose puck behind Markstrom's pad after Cody Glass' backhand caromed off a referee behind the net and back into the crease.
Backlund forced overtime when he tied it 2-2 at 8:03, one-timing a cross-ice pass from Andrew Mangiapane under Saros' right arm.
"I think there's always pressure, but our group's doing a really good job about not thinking about it and living in the moment," Sanford said. "It's so cliche ... you hear teams and players say that all the time, but it works. It's proven to work and it's working for us right now."