The rookie's third-period tally lifted the Sharks to a 3-2 win over Minnesota and eliminated the Wild from the Western Conference playoff race. Nabokov made 25 saves and improved to 12-0-1 in his last 13 starts against the Wild.
Couture, playing in his 21st NHL game, tipped Dan Boyle's shot past goalie Niklas Backstrom. That broke a tie with 3:51 remaining and sent the Sharks to their fourth victory over Minnesota this season.
"It's always good after you lose a game to come out stronger," said Nabokov, who was pulled from the Sharks' 5-1 loss in Dallas on Wednesday. "It's not easy, especially teams like that. They have nothing to lose and they play loose. That's dangerous."
Backstrom also stopped 25 shots, but it wasn't enough to keep Minnesota from missing the playoffs for the second straight year.
Manny Malhotra and Scott Nichol scored to give the Sharks a 2-0 lead in the first period. Cal Clutterbuck and Antti Miettinen answered in the second to get the Wild even. Minnesota, which won the Northwest Division two years ago, was eliminated from the playoff hunt with four games remaining.
"The writing was kind of on the wall, but it's never fun playing these games when you're mathematically eliminated," forward Andrew Brunette said. "We'll have to find a different kind of motivation and end the year on the right note."
The Wild nearly grabbed a 1-0 lead, but captain Mikko Koivu's shot during a power play hit the right post. San Jose then took over.
Malhotra redirected Rob Blake's shot past Backstrom at 7:19. The Sharks' lead doubled 5:03 later during a scramble in front. The puck squirted loose and was snapped in by Nichol, who recorded his fourth goal of the season and his first since Feb. 4.
"Get the puck on net," said Sharks coach Todd McLellan, acknowledging a lucky break on the winning goal. "Obviously, we didn't plan on Danny Boyle breaking his stick, but we needed to score an ugly goal. Get it to the net."
Minnesota rallied in the second period. Guillaume Latendresse's shot deflected in off Clutterbuck's arm 5:10 in, and the Wild tied it at 6:59 when Miettinen fired a shot past Nabokov during a power play.
The Sharks' first power play nearly produced a short-handed goal for the Wild. James Sheppard got free for a breakaway and he was hooked by Boyle on his way to the net. Sheppard was awarded a penalty shot, but Nabokov made a pad save during the attempt to keep the game tied.
On the final goal, Boyle's stick snapped as he was trying to fire a slap shot from the blue line. The slow, trickling puck got through the Minnesota defense and made its way to Couture, who tapped in his fourth career NHL goal.
"Because he broke his stick, the puck was moving so slow that guys had time to react to it," Wild coach Todd Richards said. "We had two guys going for it, and through the collision and reaching, it missed everybody. It comes out the other side, and they happened to have a player sitting right there."
Members of the NCAA champion Minnesota Duluth women's hockey team were honored before the game. The Bulldogs defeated Cornell 4-3 in three overtimes on March 21, for their fifth national title in 10 years. ... Joe Thornton, who leads the Sharks with 85 points, sat out for the third consecutive game because of a lower body injury. ... The Wild assigned D Cody Almond to Houston of the AHL. Almond played in Minnesota's 4-0 loss to Chicago on Wednesday, his third NHL appearance this season. ... Referee Dan Marouelli was hit in the face by Brent Burns' slap shot late in the second, but remained in the game. He and Burns spoke at center ice after the period ended.