Scott Nichol also scored for Nashville, which moved past Vancouver into a tie with Colorado for seventh place in the Western Conference.
Craig Conroy had the lone goal for Calgary, which stayed tied with Minnesota atop the Northwest Division after the Wild lost to Atlanta.
As Adrian Aucoin attempted to carry the puck up the boards deep in the Flames' end, Dumont knocked the puck off his stick and it went to Alexander Radulov cutting through the slot. Radulov's shot hit the goalpost, but the puck bounced to Dumont, who pounced on the empty net for his 26th goal.
Tied at 1, Calgary nearly pulled ahead with four minutes left but Cory Sarich's wrist shot from the blue line through a crowd rang off the goalpost.
The Flames got another chance to win it when Radulov got called for closing his hand on the puck with two minutes left in the third. However, the Flames' power play couldn't produce the winning goal, ending 0-for-6 with the extra man.
In the last six games, Nashville has killed of 29 of 31 penalties.
Trailing 1-0 after two periods, Nashville tied it at 8:58 of the third when Nichol burst down the wing into the Flames end and fired a wrist shot from a sharp angle that deflected off Kiprusoff's shoulder and into the top corner.
The goal ended Kiprusoff's shutout streak at exactly 115 minutes.
The only goal of the opening 40 minutes came at 11:38 of the second period off a turnover deep in the Nashville zone.
Ryan Suter's pass across the ice to Shea Weber skipped past Weber's stick and caromed sharply off the side boards and into the slot. Conroy swooped in and ripped a shot into the top corner past Dan Ellis.
