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.
Pavelec faced 37 shots in his 36th start of the season as the Jets lost for the fourth time in their past five games.
San Jose has picked up 11 of a possible 12 points in January and continue to lead the Pacific Division with 53 points.
Winchester got his stick on a bouncing puck for his fifth goal of the season.
Notes: 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.