Shots were 24-13 in Calgary's favour through 40 minutes.
The Flames had a glorious chance to extend the lead just over three minutes into the third. A Martin Pospisil entry pass hit Kadri in stride, in alone on Saros, who closed the five-hole just as the Flames forward tried to tuck it through.
Vladar saved his best work for the final frame - a pair of technical kick saves - one off Marc Del Gaizo, the second off Novak from the slot.
And his forwards helped. Coleman knelt down in front of a point blast, absorbing the puck with a wince. Then, on a Nashville powerplay, Rooney hurled himself recklessly at a slot shot, preventing a high-danger opportunity for the hosts to equalize.
At the other end, Saros made another important stop off Matt Coronato, who was set up beautifully by Kadri in the slot on a 3-on-1 rush with about five minutes left in regulation time.
Vladar and crew continued to fend off the hosts right to the final buzzer, Nashville coming close on a hit post, and again through Forsberg on a goal-mouth scramble with 26 seconds left to play.
The Flames keeper made 17 saves on the night.
With the win, Calgary improves to 14-10-5, ahead of a five-game homestand beginning Thursday night against the Lightning.