With eight seconds remaining in a tied game, Joe Pavelski won an offensive zone faceoff to Logan Couture, who set up Vlasic to snipe the winning goal past Jets goaltender Michael Hutchinson with 4.5 seconds remaining.
"To finish it off before overtime is big. It worked out for us," said Vlasic, whose goal denied at least one point to a team the Sharks are battling for playoff positioning.
"We won the draw and Couture made a nice pass. There was a little bit of room and I was able to pick a little spot up there," he added.
Brent Burns and Melker Karlsson also scored for the Sharks, while Andrew Ladd and Jay Harrison replied for the Jets.
The clubs entered the night holding both of the Western Conference's wild card playoff spots. The Sharks (21-14-5) moved into a tie with Vancouver and Los Angeles for second in the Pacific Division.
"We couldn't be happier, obviously, to get the win and fix what was broken," said Sharks coach Todd McLellan, whose team entered the night with one win in its past five games. "We didn't play very well against St. Louis the other night and Winnipeg plays a similar game."
The Jets (20-13-7) remained winless in 13 games in which they've trailed after two periods, but coach Paul Maurice said it isn't due to his team's approach in the third.
"We're not holding back," said Maurice. "I liked our third period as much as any part of our game tonight. I don't think we can open up our game more to generate more."
Maurice said his biggest concern was the first 30 minutes of the game, in which his team didn't generate much offence at even strength.
"You have eight seconds left and you want to get it to overtime. So, yeah, it stings," added Ladd, Winnipeg's captain.
Harrison's shot from the high slot on the power play got through Antti Niemi's pads at 4:03 of the third period to tie the game at 2-2.
"We were hanging around long enough to give ourselves a chance of winning the game, and those are stingers, ones you lose at the end like that," said Harrison.
The Jets needed to kill two penalties in the final 10 minutes as Ladd was called for illegal contact to Karlsson's head and Paul Postma took a delay-of-game minor. All told, San Jose had seven power plays to Winnipeg's three.
Hutchinson, who entered the game leading the league in save percentage and goals-against average, stopped 34 of 37 shots. Making his fifth start in six games, Hutchinson continued to see an increased workload at the expense of longtime starter Ondrej Pavelec.
"Their team does a good job of getting a lot of traffic and shooting for sticks," said Hutchinson. "There's very few shots tonight that came off their guy's sticks clean and made it all the way to me without being touched."
Niemi made 17 saves on 19 shots in his eighth start in nine games.
The teams traded power-play goals in the first period, with Burns opening the scoring at 2:53. Pavelski won an offensive zone faceoff, and two passes later Burns hammered home a slap shot from the point for his 11th goal of the season.
"It's a fast team and a tough building to play in," said Burns. "I think it was going back and forth. It was good to get that last one."
Ladd took a pass in the high slot from Bryan Little and beat Niemi to his glove side for his 15th goal of the season at 9:17.
The Sharks had a 13-4 advantage in shots after the first period, limiting the Jets to only one even-strength shot.
Pavelski floated a shot from the right boards that Karlsson tipped down and through Hutchinson's pads to give the Sharks a 2-1 lead early in the second period.
The Jets begin a three-game West Coast road swing Thursday in Arizona, while the Sharks continue their three-game trip Tuesday in Minnesota.
Notes: San Jose centre Joe Thornton missed a second straight game with an upper-body injury. Thornton was placed on the injured list after playing in 319 consecutive games. … Grant Clitsome missed the game with an undisclosed injury, making it five Winnipeg defencemen on the shelf. Call-up Julien Brouillette made his first start as a Jet. … Winnipeg forward Michael Frolik went to the dressing room following a hard check late in the second period, but returned at the start of the third.