Handzus tallied for San Jose in the second round and Burns put the game away in the third round with a goal that gave the Sharks their seventh win in nine games.
Both Handzus and Burns had assisted on Torrey Mitchell's backhander that tied the score 1-1 at 10:44 of the third period.
Chris Butler scored in the second period for the Flames, who have lost their last six road games. Miikka Kiprusoff stopped 31-of-30 shots.
Antti Niemi made 27 saves in goal for San Jose.
Notable deaths of 2014 [PHOTOS]