Killorn scored an empty-net goal at 18:09 for the 3-1 final.
Gauthier assisted on both power-play goals for his first multipoint game in the NHL. He was briefly benched for poor play in Anaheim's home opener against the Utah Hockey Club on Oct. 16, a 5-4 overtime win.
"I'm capable of it, it's just knowing that," Gauthier said. "Just playing the last couple games, I felt like I took a step in the right direction each game and, obviously, it was nice to get on the scoresheet to score for it. The most important thing is we got the job done and we won."
The Sharks had the best scoring opportunity in the scoreless first period.
Danil Gushchin's shot off the rush clipped Dostal's arm at 6:47, hit the crossbar, came down on the goal line and then slid off the post before Anaheim defenseman Brian Dumoulin swept the puck out of the crease.
"You've got to be lucky sometimes, too," Dostal said.
After the Ducks took a 1-0 lead, the Sharks hit the post again on a deflection by Barclay Goodrow before Dostal covered the puck with 16 seconds left.
"We just have to stick together as a team, pick each other up off the ice," San Jose defenseman Cody Ceci said. "It's tough what we're going through. We are competing hard, we're working hard, just not getting the results. It's results-based, so we need to get a win. We need to find a way to get it and just be there for each other until we get it."
NOTES: Sharks defenseman Jack Thompson assisted on Granlund's goal for his first NHL point. ... Terry has four goals in his past five games, and extended his point streak to five (four goals, one assist). ... Ducks defenseman Pavel Mintyukov returned after missing the 4-1 loss to the Los Angeles Kings on Sunday (illness) and played 18:07. ... Ducks center Isac Lundestrom returned after missing the past two games with an upper-body injury and had three shots on goal in 14:52 of ice time.