Calgary had a 2-1 lead with the first period winding down when Fisher put a wrist shot by Miikka Kiprusoff to start Nashville's barrage.
He then kicked off a stretch of three goals in a span of 2:30 of the second period with Nick Spaling and Brandon Yip also scoring during the outburst to give the Predators a 5-2 advantage.
Jarome Iginla scored for Calgary with a minute left.
Nashville ended a four-game losing streak with Pekka Rinne stopping 19 shots. Kiprusoff allowed all the Nashville goals on 23 shots. The Flames have lost two straight.