Keith Kinkaid made 37 saves for the Devils (3-0-0), who denied the Sharks on three power-play opportunities. Kinkaid made five saves in the final 1:50 during a too-many-men minor when San Jose held a 6-on-4 advantage.
"Everybody was sacrificing their bodies; the Sharks are a shoot-mentality team so they throw everything to the net," Kinkaid said. "I think we did a good job systematically and I just had to make the saves."
The Devils were outshot 39-36, but held an 18-10 advantage in the third.
"Keith's played great and he's made some big timely saves," said Palmieri, who has a three-game goal streak. "He's kept us in games or given us momentum when maybe things were getting sloppy in our own end. He's done more than you can ask of a starting goalie, and hopefully he continues to make those saves."
Palmieri scored on a two-man advantage to tie the game 1-1 at 7:06 into the second. Taylor Hall (two assists) sent a pass from the right-wing boards into the left face-off circle to Palmieri, who became the fourth player in NHL history to score multiple goals in at least three consecutive games to open a season, joining Patrick Marleau (four games in 2012-13), Peter Stastny (three games, 1982-83) and Cy Denneny (three games, 1917-18).