Matt Coronato also scored and Dustin Wolf had 29 saves for Calgary, which won for the seventh time in 11 games (7-2-2).
Will Smith scored twice and Macklin Celebrini had two assists for San Jose, which has lost six straight (0-5-1). Georgi Romanov finished with 29 saves.
Klapka tied the score 1-1 at 8:12 of the third as he knocked in the rebound of Nazem Kadri’s shot on a 2-on-1 rush.
Frost put the Flames ahead with a power-play goal with nine minutes remaining as he beat a screened Romanov with a shot from the left faceoff dot.
Coronato made it 3-1 with his 22nd of the season with 4 1/2 minutes to go, and Smith got his second of the night and 16th of the season with 2 minutes left.
Smith gave the Sharks a 1-0 lead as he got a pass from Nikolai Kovalenko from behind the goal line and quickly beat Wolf from in front with 25 seconds left in the first for his 15th.
Flames: Calgary trails Minnesota by four points for the second wild card in the Western Conference with five games remaining — one more than the Wild.
Sharks: The league-worst Sharks are just 6-21-4 over their last 31 games.
Wolf made two saves on two quick shots from Celebrini in the second period that kept the Sharks from adding to their 1-0 lead. The second save was with Wolf sprawled out on the ice.
The Sharks, averaging 27.0 shots per game, finished with 31 — including an 18-8 advantage in the first period.
Flames visit Anaheim on Wednesday, and Sharks play at Minnesota.