San Jose coach Bob Boughner, forwards Andrew Cogliano, Jonathan Dahlen and Matt Nieto, and defensemen Erik Karlsson, Jake Middleton, Radim Simek and Marc-Edouard Vlasic were unavailable, as was forward and captain Logan Couture, who has a non-COVID illness.
Timo Meier scored a power-play goal at 1:07 of overtime after Kyle Connor tied it 1-1 at 16:48 of the third period.
"It was a tremendous effort," said Sharks assistant coach John MacLean, who filled in for Boughner. "[The players] had great energy and they all knew they had to play."
Meier scored from the right circle after Nikolaj Ehlers was called for interference on Sharks defenseman Brent Burns.
"It [stinks] when that happens close to the end," Burns said of the tie. "But it was good to get the two points right after that."
Tomas Hertl scored, and James Reimer made 34 saves for San Jose (5-3-0), which had lost three straight.
"That's what adversity does, right?" Reimer said. "It forces you to depend on each other and bring out the best in each other."