The Sharks surrendered a 2-1 lead in the latter stages of the second period and appeared headed for a setback before Clowe saved the game with his 18th goal of the season.
He then beat Mikka Kiprusoff in the third round of the shootout to bring the Sharks their sixth straight win. San Jose extended its lead in the Pacific Division to three points over Phoenix.
The Sharks finished their February schedule with an 11-2 record.
David Moss and Niklas Hagman scored for the Flames over the final 4:06 of the second period to give them a 3-2 advantage heading into the third.
Antti Niemi made 21 saves to get the victory. Kiprusoff stopped 30 shots for the Flames, who have lost four out of seven.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged