San Jose (44-25-9) moved to within six points of the Calgary Flames for first place in the Pacific Division and is six points ahead of third-place Vegas. The Sharks, who host Calgary on Sunday, can finish no worse than second.
"We needed a win to solidify home ice," coach Peter DeBoer said. "We needed a win mentally because we haven't found a way lately, although I think our play hasn't been as bad as our record. But there's no corners turned or any dramatic changes. We've played good hockey all year, and I think that's going to continue."
WATCH: [All Golden Knights vs. Sharks highlights]
The Golden Knights (42-30-7), who clinched a playoff berth Friday, have lost five games in a row, the longest losing streak in their two seasons in the NHL.
"We're going into the playoffs, it's time for us to step it up," said William Karlsson, who scored for Vegas. "We're losing too much. Things haven't been going our way, but that's in the past. We have to look forward."
Brent Burns scored a power-play goal from above the left face-off circle 22 seconds into overtime.
Tomas Hertl and Evander Kane scored, and Martin Jones made 23 saves for the Sharks.