Evander Kane had a goal and an assist, and Devan Dubnyk made 28 saves for his first win with the Sharks (7-7-2), who lost to the Blues 3-2 in overtime Thursday.
San Jose recovered after surrendering a three-goal lead.
"We had to control our emotions," Sharks coach Bob Boughner said. "We stuck with it. We had some great character (penalty) kills out there, and we got three points out of this building and it could easily have been four."
Jordan Kyrou had a goal and two assists, and Ryan O'Reilly and Brayden Schenn each had a goal and an assist for the Blues (10-6-2).
"We definitely scored enough to win the game, but too many mistakes defensively," St. Louis coach Craig Berube said. "We've just got to tighten that up. Got to be better.
"We had a couple turnovers, and they countered off of them. One was a goalie handle, and it was unfortunate, but just in general, a couple turnovers went in our net, and it's stuff that we know what we're supposed to do and know where we're supposed to be. When you turn the puck over and you make the wrong play with the puck, people counter off of it, and they did a good job of that and they scored some goals."
Couture scored unassisted to give San Jose a 5-4 lead at 11:51 when his shot from the right face-off circle went in off the back of goalie Jordan Binnington.
"I'm counted on to score goals and produce offensively," Couture said. "I want to score and help our team. We had a lot of guys step up tonight. Different guys played well. The list goes on. We scored five goals. We've had a tough time scoring goals this season. When we play like that, we'll find the back of the net."
After Patrick Marleau gave the Sharks a 4-1 lead at 8:18 of the second period with his first goal of the season, the Blues scored three straight before the end of the period.