"I guess if you're on the bench you're more a spectator than kind of into the game," Hiller said. "Sure you follow it and you kind of feel the up and down of the team and everything. I was kind of prepared that we'd go to overtime, anyway, but yeah, I was worried because I saw we took two penalties there in a row and then suddenly the next thing that I know I was on the ice.
"It doesn't give you a whole lot of time but it definitely helps to have had a lot of games under my belt and having some experience to come in and just focus on the way I want to play."
Video: CGY@SJS: Bennett steals puck, scores to open scoring
Hiller stopped all five shots he faced in regulation and three more in overtime. In the shootout, he gave up a goal to Logan Couture in the first round before stopping Joonas Donskoi, Joe Pavelski and Patrick Marleau.
Jiri Hudler's goal in the fourth round gave the Flames the win. Calgary's Joe Colborne scored in the first round, beating Alex Stalock, after Couture had scored against Hiller.
"He got us the win," Flames coach Bob Hartley said. "Our penalty-killers were unbelievable, but our best penalty-killer was [Hiller], coming right off the bench. He was outstanding. I don't know, what did he play, like three minutes? He got the player of the game."
Sam Bennett, Mark Giordano, Sean Monahan, Mikael Backlund and Kris Russell scored in regulation for the Flames (25-25-3), who won their third in a row.
Video: CGY@SJS: Russell puts home rebound on PP to tie it
The Sharks (28-20-5), who overcame a 4-1 deficit after one period but couldn't hold a 5-4 third-period lead, got goals from Tommy Wingels, Couture, Marleau, Donskoi and Dylan DeMelo.
Just before Hiller entered the game, Bennett went to the box for elbowing Paul Martin, and Giordano joined him after tripping Donskoi, who piled into Ramo as he went headfirst into the crease. Ramo had to be helped off the ice.
The Sharks had four shots during the two-man advantage, but Hiller stopped them all.
"Kudos to him," Couture said of Hiller. "He stepped in on a 5-on-3 and made unbelievable saves. He saved them a couple of points."
Ramo made 29 saves on 34 shots. Hartley didn't disclose Ramo's injury but said "it doesn't look very good" for him now. Hiller will start against the Arizona Coyotes on Friday at Gila River Arena.
Flames forward David Jones also left the game with an undisclosed injury; Hartley said he had "no idea of the extent of the damage."
Stalock, making his first start since Jan. 18, gave up five goals on 22 shots.
"He hadn't played in a long time," Sharks coach Peter DeBoer said of his backup goaltender. "I actually thought after the first period he settled in. Made a couple of big saves to give us a chance to get to overtime. I expected him to be a little rusty. We were a little rusty around him in the first period, and I thought in the second and third he was really good."
The Sharks entered the third period trailing 4-3 but pulled even when Donskoi, a rookie forward, scored his ninth goal of the season at 2:46. He took a cross-ice pass from Couture and beat Ramo from close range.
Video: CGY@SJS: Donskoi puts it home to tie the game in 3rd
DeMelo, a rookie defenseman, gave San Jose its first lead at 5:55 with the Sharks' third power-play goal of the game. Joel Ward sent a cross-ice pass to the left circle, and DeMelo put the puck past Ramo for his second goal of the season.
But Calgary tied it when Russell scored a power-play goal at 8:08. Bennett fired a sharp-angled shot that Stalock stopped but couldn't control, and Russell knocked in the rebound from close range.
Couture hit the post just over a minute into overtime. Marleau had a good scoring chance with two minutes left, but Hiller made the save, and also stopped Justin Braun's shot from close range with 1:36 remaining.
The Sharks went 3-for-9 on the power play; Calgary was 2-for-5.
Flames forwards Monahan, Johnny Gaudreau and Lance Bouma returned the lineup after being scratched Tuesday against the Toronto Maple Leafs as punishment for arriving late for practice Monday. Gaudreau had two assists; Monahan scored his 17th goal of the season.
Video: CGY@SJS: Sharks play tic-tac-toe to extend lead
The Flames raced to a 4-1 first-period lead, scoring four times on their first eight shots. The four goals were the most San Jose has allowed in a period this season.
Bennett put Calgary ahead at 4:12 with an unassisted goal. He intercepted Braun's pass in the neutral zone and beat Stalock from above the right circle with a wrist shot for his 15th of the season. Giordano made it 2-0 at 5:18 with a power-play goal, a snap shot from the left circle that deflected off Braun's stick and past Stalock for his 13th of the season.
"You've got to be ready," Braun said. "I was one of the guys who wasn't ready at the drop of the buck, didn't have my A-game that first 20 [minutes], and it showed on the scoreboard."
San Jose got one goal back at 8:45 when Wingels scored his sixth of the season, converting a pass from Donskoi.
Video: CGY@SJS: Marleau chips it through on PP to cut lead
But Monahan scored on a 4-on-2 rush at 12:46, and Backlund made it 4-1 by beating Stalock on a breakaway at 13:26, extending his goal streak to three games.
Couture cut the deficit to 4-2 with a power-play goal 48 seconds into the second period, and Marleau scored during a 5-on-3 advantage at 11:17.
"We knew that they would come at us," Hartley said of the Sharks. "Even though we were leading 4-1 after one, there was way too much time for that powerhouse. You look at their offensive power. We knew they would get lots of pressure on us, and that's what they did. But a gutsy effort from our guys. I'm very proud of our team."
Sharks defenseman Brenden Dillon missed the game with an upper-body injury. Matt Tennyson replaced him and played for the first time since Dec. 8.
Video: Flames defeat Sharks in shootout