Hill preserved the lead by making a glove save on Johnny Gaudreau's wrist shot from the slot at 15:18 before stopping Tyler Pitlick's redirection at 17:32.
"Tremendous," Couture said. "He played great. Big saves. A few times they were off tips. There were a few times the puck bounced around, but he was a rock for us tonight."
Tomas Hertl (18:57) and Jonathan Dahlen (19:19) each shot into an empty net for the 4-1 final.
"I think they came out and we had the better of them in the second, and I'm sure we thought we could continue that, and they came out with a little more jump (in the third)," Calgary coach Darryl Sutter said. "Who scored the winner there, was it Couture? Logan and Burns. … Those guys played a lot and were probably better than our top guys tonight."
Barabanov scored 47 seconds into the second period to give the Sharks a 1-0 lead. He shot five-hole on Markstrom after receiving a stretch pass from Burns.
Mangiapane responded 30 seconds later to tie it 1-1, jamming in a rebound at the right post.
Calgary outshot San Jose 27-12 through two periods.
"They get a bounce in the third and were able to shut us down for the rest of the third," Lucic said. "It [stinks] losing that one because you know you play so hard in the second and only get one out of there, and sometimes when you let teams hang around that's what happens."
NOTES:Hill had two assists in the game, his first two NHL points. He is the second Sharks goaltender to record two assists in a game, joining Vesa Toskala (Feb. 3, 2007). … Burns has eight points (one goal, seven assists) in a five-game point streak. … The Flames are 0-2-1 against the Pacific Division and 0-2-3 against Western Conference opponents.