Shesterkin was injured when he went down on his stomach in the crease as the puck was around the blue paint. He couldn't get up under his own power and didn't put weight on his right leg as he was being helped off the ice and down the tunnel to the Rangers' dressing room.
He is 13-3-2 with a 2.05 goals-against average and .937 save percentage in 18 games this season, including 8-1-0 with a 1.76 GAA and .945 save percentage (14 goals allowed on 256 shots) in nine games since Nov. 8.
"I think we've grown a lot in the last year or two and we're able to handle these things no matter who goes down," Strome said. "We've had a ton of games where we've played with 11 forwards because guys get hurt or whatever. I think guys responded well on the bench, just tightened up and made sure we helped him out."
Georgiev hadn't played since allowing four goals on 18 shots in two periods against the Buffalo Sabres on Nov. 21. Shesterkin relieved him for the third period and the Rangers won 5-4.
The Rangers already had a plan for Georgiev to start against the Chicago Blackhawks at Madison Square Garden on Saturday (8 p.m. ET; MSG, NBCSCH+, ESPN+, NHL LIVE).
"He's a good goaltender and he's been waiting for an opportunity," Strome said. "He went in there and finished her off and he'll probably get in there tomorrow. That's what good teams do, next man up. It was a good example of that tonight."
The Sharks have been shut out three times this season and held to two or fewer goals in nine of 12 games since Nov. 11.
"Our goalie played fantastic, made some huge saves, but if we can find that goal it's a different game," Sharks forward Timo Meier said. "I thought defensively we did a good job; it's just that offensive stuff, that extra inch to get in front of the net, if we can get that we're going to have success."
Strome gave the Rangers a 1-0 lead at 6:03 of the first period on a slap shot from inside the right circle.
The forward started the play from the right corner but came back up the zone to the top of the right circle, where he received the puck from Adam Fox. Strome had space to skate into the circle before getting off his slap shot that beat Hill on the near side.
He said he moved up higher in the zone by design to make himself a shooting threat.
"If I don't climb up higher, I'm just kind of a passing option, which works sometimes but a lot of teams are able to take it away," Strome said regarding his fourth goal of the season and first on the power play. "The way [the Sharks] penalty kill, they're very good in the middle, they don't give you too much space. You've got to use the outside so just find the good ice and let one go. Nice to get one."
The Sharks struggled to handle the Rangers pressure late in the third period, turning over the puck as they attempted to get it out of their zone to get Hill off the ice for the extra skater.
"They did a good job pressuring and guys got tired, got caught," Meier said. "When you're down a goal and it's 1:30 left you want to get in the O-zone, get fresh guys on and try to put yourself in a spot to score. They did a good job putting pucks behind us and making it hard for us."
NOTES: It was the second shared shutout in Rangers history (Dec. 15, 2016 at the Dallas Stars, Henrik Lundqvist and Antti Raanta). … Fox's assist on Strome's goal was his seventh in a five-game point scoring streak and his 19th of the season, the most among NHL defensemen.
