Gaudette tipped a hard pass by Stutzle past Sharks goaltender Mackenzie Blackwood with 4:12 left for his fourth goal in as many games. San Jose had tied the game earlier in the period on goals by Macklin Celebrini and Will Smith.
Stutzle’s assist was his third of the game.
Ottawa scored twice on its first three shots to lead 2-0 after one period. Stutzle had assists on both, scored by Gaudette at even strength and Josh Norris on the power play.
Mario Ferraro also scored for San Jose. Tyler Kleven recorded his first NHL goal for the Senators.
The Senators have won consecutive games after losing five straight, while the Sharks have lost four of five.
Sharks’ leading scorer Mikael Granlund missed his second straight game (upper body).
Senators: Gaudette, who recently moved to the top line, has 11 goals on 31 shots this season. He entered the game second in the league behind Alex Ovechkin in goals per 60 minutes (2.61).
Sharks: The Sharks outshot the Senators 39-11, but Ullmark, who improved to 6-0 in his career against San Jose, was solid in net after losing his last four starts.
The Sharks had all the momentum after rallying from two goals down in the third, but Stutzle and Gaudette combined to give Ottawa the lead with 4:12 to go.
The 22-year-old Stutzle has 28 points through 21 games, leading the league in points among players under the age of 23.
The Sharks host the Kraken on Friday, while the Senators play at the Kings on Saturday.