Couture scored at 13:36 of the second period and left wing Winchester made it 2-0 early in the third for the Sharks, who have won five of six.
Couture scored for the second straight game, but it was defenseman Brent Burns that was key on the play.
Burns picked the puck up outside his blue line, bounced the puck off the left-side boards and then took control of it again.
Just as it looked as if he was going to go around the back of the net, Burns sent a quick backhand pass to Couture, who snapped the puck into the net while goalie Ondrej Pavelec was out of position.
It was Couture's 18th goal of the season.
"I got a little bounce off the glass there," Burns said. "Just got the forecheck going.
"With the way the breakout happened, the forecheck kind of stuck because I lugged it up and so, kind of just the way it happened."
Niemi was impressed as he watched the play.
"(Burns has) got the speed and he saw there's a chance to go all the way and he was able to make a good pass," Niemi said.
Winchester got his stick on a bouncing puck for his fifth goal of the season.
Winnipeg Jets coach Claude Noel is worried about his team's mental state.
"For me, it's going to be more about dealing with the morale of our team," Noel said.
"The team isn't going to be feeling very good right now and that's probably one of more of the issues we're going to have to deal with."
Winnipeg's players were down on themselves for the effort they gave in the loss.
Pavelec, who faced 37 shots, said they "didn't deserve to win."
"It was a tough game tonight against a very good team," Pavelec said. "They did everything to win the game."
Jets captain Andrew Ladd said it's time to learn from their mistakes.
"How many times have we stressed about getting pucks deep and making sure that we're doing those smart things and how many turnovers did we have tonight? Way too many," Ladd said. "At some point we gotta figure out if we want to be a playoff team or if we're just content with being this win one here, win one there.
"Personally, I'm sick of having these little stretches."
Forward Nik Antropov said the Sharks' skaters and Niemi played well, but he and his teammates didn't do enough to help themselves.
"(San Jose) played a good game, but it's up to us," Antropov said. "I don't think we pushed hard enough. The third period was just unacceptable. It was a flat period, nothing was going on.
"We've got to blame ourselves, nobody else."
The Sharks are 17-1-3 when leading after the second period, while the Jets are 0-14-2 when trailing after two ... Jets defenseman Dustin Byfuglien, selected to the NHL All-Star game, missed his eighth game with a knee injury. Fellow defenseman Zach Bogosian was out with a lower-body injury he suffered in Tuesday's loss to Boston.
Copyright by STATS LLC and The Associated Press
Arena: MTS Centre Location: Winnipeg, Manitoba
Referees: Dan O'Rourke, Kevin Pollock Linesmen: David Brisebois, Pierre Champoux