Jonathan Huberdeau was the first shooter and scored, but Cody Glass - who had to beat Jacob Markstrom as Nashville's third shooter to keep his team alive - put one under the Calgary's 'tender's glove.
Mikael Backlund was stopped and then Tommy Novak scored to end it.
The Flames finished with a 45-29 edge in shots, while they went 0-4 on the powerplay.
Nashville was 0-6 with the man advantage.
The Flames had three glorious chances in the OT: first Huberdeau fed MacKenzie Weegar in tight that Juuse Saros stopped, then Rasmus Andersson tried a wraparound on Saros but it got deflected wide, finally, Tyler Toffoli stepped into a point-blank range slapshot that the Preds netminder stopped.
Troy Stecher and Backlund scored in regulation, while Andrew Mangiapane had a pair of helpers.
Calgary trailed 1-0 and 2-1 in the outing before forcing extra time.
The Flames wrap the season up at home Wednesday against the San Jose Sharks.
Markstrom made an stellar desperation save early, laid out flat on his side, stretched to the far post and able to get his arm down on an attempt on a Egor Afanasyev pass attempt to former Flame Mark Jankowski who was looking for a backdoor tip-in.
Calgary's first good look came courtesy of a Noah Hanifin point blast that Saros stopped but left a rebound that Backlund fired over the net from the doorstep.
Calgary got another big stop from Markstrom, who kicked out his left pad to deny Colton Sissons who was streaking in all alone.
But a misplayed puck behind led to the opening tally, Afanasyev getting the puck alone out front with an empty net and snapping it in at 5:49.
Blake Coleman was sprung early in the third, grabbing a clear and then going inside out on Spencer Stastney for a shot which was stopped, then slipping away from him to grab the rebound with his backhand, before going forehand in tight but was again denied.
Stecher tied it up at with absolute gem at 7:41, his third of the season, all in Flames' silks since joining the team in a trade with the Coyotes. Dillon Dube skated the puck into the Predators zone, passing down to Mangiapane who took the puck off his skate to his stick and found Stecher up high skating into the zone, and he put a perfectly-placed wrister farside past a screened Saros.
