"Desperation time was a while ago, so yeah, this is where pride has to come in," Blues coach Mike Yeo said. "We have to make sure that we obviously stop the bleeding. We have to start building confidence in the game that we're playing and have a feeling that showing up to the rink that everybody's going to be ready to go.
"We can't just accept that we're playing good and not winning hockey games. That's not good enough. We need points, we need wins and we need to start doing it now."
All five goals in the game were scored in the second period.
Couture gave the Sharks a 1-0 lead at 6:35 when he lifted a rebound over Hutton after an initial shot from the slot by Boedker.
It was Couture's 24th goal of the season.
"It's been a good year so far offensively," Couture said. "I think I've capitalized on the chances I've had. Tonight was a nice one to get the puck on my stick and a wide open net, so hopefully I can continue it."
Barbashev tied it 1-1 at 8:23 on a wrist shot from the right face-off circle that beat Jones high on the short side.
Joonas Donskoi put the Sharks ahead 2-1 at 12:05 on a quick snap shot from the left face-off circle before Boedker made it 3-1 at 15:54 on a shot from close range, his fifth goal in four games.