Nieto made it 3-2 when Alexander Barabanov went hard on the forecheck to win the puck behind the net and passed to Logan Couture, who fed Nieto at the edge of the crease.
"The face-off is kind of a broken play and Barabanov gets in and creates a turnover, Logan works to a good scoring position, and I was able to beat my check to the net," Nieto said.
The only faster goal to start a period in San Jose history was by Stephane Matteau (eight seconds into first period) on Jan. 11, 2000.
"They're a heavy team around the net, that's their strength, and I think we did a good job for the big part of the game," Arizona coach Andre Tourigny said. "But in the end, that's the kind of stuff that's tough to defend. We made a big mistake of starting the third with a little bit of a casual play, and they capitalized on it by getting their stick open."
Jaycob Megna scored at 14:10 for the 4-2 final on a one-timer from above the circles off a face-off win. It was his first goal in 46 games.
"To start the third, we get the goal early and I thought we did a good job for the most part of not giving them much after that," Sharks coach David Quinn said. "We got the big insurance goal we needed, and Kaapo was good in net."
"The third period, we kind of had them running around a little bit and took over with our momentum," Labanc said.