Pavel Dorofeyev and Brett Howden scored, and Jack Eichel had a goal and an assist for the Golden Knights (1-0-1). Mark Stone had two assists, and Akira Schmid made 20 saves.
Eichel's goal tied the game 3-3 at 18:26 on a dump-in from center ice. Eichel lifted the puck into the Sharks zone, where it landed and then bounced awkwardly by Nedeljkovic.
"Just got caught in between on the third one, whether to go out and try to catch it in the air or let it bounce and hop," Nedeljkovic said. "Usually when you're in between at this level, bad things happen."
The game-tying goal came seconds after San Jose's William Eklund was twice denied a chance to score into the empty net.
"Well, we were certainly fortunate. The ironic thing is, we played that goalie twice in preseason. He's played great against us, handled the puck well," Golden Knights coach Bruce Cassidy said, "A little bit of October hockey, I guess. The hockey gods went our way tonight. I actually feel bad for him. He played well in there."
Jeff Skinner, Alexander Wennberg, and Philipp Kurashev scored for the Sharks in their season opener. Shakir Mukhamadullin had two assists, and Nedeljkovic made 27 saves.
"I liked our game. We were skating. Playing with good structure for the most part," Sharks coach Ryan Warsofsky said. "Obviously, a couple of things we've got to clean up, but I liked our game. Our guys competed."
Skinner gave the Sharks a 1-0 lead at 5:31 of the first period. After Ty Dellandrea's shot deflected off traffic in front, Skinner chipped the rebound in.