Nick Bonino tied it 1-1 on a tip-in from the slot off a pass from Jonathan Dahlen at 9:10 of the first, one second after a Sharks power play expired.
"We have to grind teams down, grind wins out," San Jose coach Bob Boughner said. "We have to do it as a team, and we have to do it a certain way. Managing the puck, managing leads and being in tight games ... we're a lot more comfortable in those games than we have been in the past couple years."
Lee, who missed the previous four games in COVID-19 protocol, hit the crossbar on a backhand from the left circle with 48 seconds remaining in the first.
"I felt good tonight," Lee said. "I had my legs, I think our line (Lee, Barzal, Josh Bailey) started off really well and we felt good tonight. One of those nights when the chances were coming and we felt that all we needed was one more opportunity, one more puck and we were going to pop one. We just didn't get that, but it's a game us three can build off of and the team as a whole can too."
Sorokin made 15 saves in the second period.
"I think that everybody's helping each other out and we're battling for each other and we're doing what we feel we have to do to give ourselves a chance to win at the end of the game and today was a perfect example of that," Karlsson said. "I think we just stuck with it, weren't getting frustrated and we just kept putting the puck in and creating and taking what we got.
"We weren't concerned how long it was going to take. We knew that by the end of the third we were going to have a chance to win the game and get it done in overtime."
NOTES: Sharks forward Jonah Gadjovich left the game after he blocked a shot with the inside of his left foot from Islanders defenseman Noah Dobson at 3:11 of the first. No update was provided after the game. ... Bailey, who missed six games in COVID-19 protocol, did not have a shot on goal in 19:11. Defensemen Adam Pelech and Andy Greene also returned for the Islanders after missing four games; Pelech had three shots, three hits and led them in ice time (24:14). Greene had an assist and blocked two shots in 19:55. … New York went 3-for-3 on the penalty kill and is 11-for-12 in its past four games.