Calgary has lost in each of its last 12 trips to Anaheim and has dropped eight of its last 10 on the road overall.
Ryan Getzlaf scored for Anaheim and Alex Tanguay answered for Calgary in the first round. After both shooters missed in the second round, Perry beat Henrik Karlsson with what proved to be the winning goal.
Rene Bourque could not keep the shootout going for the Flames.
Anaheim had dropped five of its previous seven home games. Hiller made 37 saves and Karlsson stopped 23 shots.
Swim Week Miami Beach 2014 [PHOTOS]