The Flames trailed 1-0 heading into the third period, and Matt Stajan and Cory Sarich scored to help Calgary force overtime.
In the shootout, Bourque slipped the puck under Pekka Rinne's leg to start. Alex Tanguay lost the puck, then Rinne stopped Olli Jokinen. But Miikka Kiprusoff stopped Martin Erat and Cody Franson.
David Legwand had a last chance to extend the shootout but couldn't solve Kiprusoff.
Patric Hornqvist had a power-play goal for Nashville. Martin Erat also scored and Sergei Kostitsyn added two assists.
Calgary now has gone past regulation six times in the past 11 games and improved to 7-4 in shootouts this season. More importantly, the Flames continue their rally in the Western Conference, having grabbed at least a point in 10 of the past 11 games.
It felt as if the Predators were playing for the shootout with a 6-2 record in them coming into the game. Rinne, the NHL's third star for January, got Nashville to the shootout by stopping four shots in overtime, with his best a stick save on Jokinen with 2:20 left and then smothering a slap shot by Stajan 15 seconds later.
The oh-so-streaky Predators now have lost three straight, picking up where they left off going into the All-Star weekend.
They seemed in control most of the game. Erat scored his seventh from the slot, tipping in a nice pass from Kostitsyn from the left circle at 7:43 of the first past Kiprusoff's glove. The Predators wound up taking 10 of the next 12 shots.
Kiprusoff also put Nashville on the power play at the end of the second period, swiping his glove across Predators center Nick Spaling's face. But Nashville couldn't convert.
Stajan tied it up at 1-1 at 2:57 in the third with his fourth goal of the season with a slap shot off an assist from Tim Jackman.
Nashville took the lead back with a rare goal from one of the NHL's worst teams on the man advantage. The Predators came in ranked 28th. Hornqvist scored his 16th this season with a wrister from the slot off a pass from Shea Weber, beating Kiprusoff glove-side at 4:56 — just 28 seconds after Jokinen went to the penalty box for tripping.
It didn't last long as Sarich scored from the right circle at 9:04 to tie it up at 2-all.
Notes: Erat's goal also was the 130th of his career. ... Flames defenseman Steve Staios (ribs) was a scratch. He is day to day. ... Nashville improved to 19-3-4 when scoring first and 16-3-4 when leading after the first period. ... Kostitsyn has 24 points in 28 games.