Rutta scored into an empty net on a shot the length of the ice with 21 seconds remaining for the 2-0 final.
Seattle outshot San Jose 18-8 through the first two periods.
"We played a hard hockey game," Seattle coach Dave Hakstol said. "They defend really hard. They play hard, and we knew that was the case. We had three or four really good opportunities each of the first two periods and weren't able to get that first goal. They found that first goal on a seeing-eye shot. I don't think Joey probably had a chance to see that; it went through two layers of a screen. And that's the difference in the hockey game."
The Sharks played without Tomas Hertl, who is day to day with an undisclosed injury. With Hertl unavailable, William Eklund played center for the first time at the NHL level.
"I felt like, in the first period, I was feeling it out a little bit," Eklund said. "Getting into those rush positions, especially in the [defensive] zone, you have to be a little more responsible. I felt like it got better and better throughout the game."
NOTES: Vlasic scored his 13th career game-winning goal, passing Dan Boyle (12) for the second most by a defenseman in Sharks franchise history. … Vlasic became the fifth Sharks defenseman age 36 or older to score in consecutive games. The others were Boyle (three times), Gary Suter (three), Brent Burns (twice) and Rob Blake (once). … Vince Dunn, who has 136 points (29 goals, 107 assists) during his career with the Kraken, played his 200th game with Seattle to become the sixth player to reach the milestone with the franchise. Jared McCann is the only player with more points for the Kraken with 153 (86 goals, 67 assists) at the time of their 200th game. … Seattle played its final game before the NHL All-Star break.