After a scoreless first period, the teams combined for five goals in the second.
San Jose took a 1-0 lead when Couture knocked in a loose puck at 6:35. Boedker fired in a shot from the point that goalie Carter Hutton blocked but couldn't corral. Couture was on the spot and scored his 24th of the season.
St. Louis answered at 8:23 when Barbashev scored on a wrist shot from the top of the right circle that sailed over Jones' arm.
Donskoi beat Hutton on a wrist shot from the left circle at 12:05. Couture assisted on Donskoi's first goal in 17 games.
Boedker gave San Jose a 3-1 lead at 15:54 when he scored from the left side of the net. Boedker has five goals in his last four games.
Tarasenko cut the margin to 3-2 on a power play when he fired in the rebound of a shot by Vince Dunn at 19:36 for his 24th of the season. Tarasenko has 10 points in his last nine games (four goals, six assists) against the Sharks.
San Jose's Brent Burns prevented a goal with less than five minutes left. During a scramble in front of the net, the puck hit the post and was sliding along the goal line when Burns knocked it out.
This was the first meeting between St. Louis and San Jose this season. The Blues swept the three-game season series last year, outscoring the Sharks 11-3 overall, and had won four straight against the Sharks.
Hutton made his second start in the last five games and stopped 20 shots. Previously against the Sharks, he was 4-0-1 with a 0.98 goals-against average, a .968 save percentage and two shutouts in five career starts.
NOTES: The Sharks acquired C Eric Fehr from Toronto for a seventh-round pick in the 2020 draft. ... Blues C Vladimir Sobotka was out of the lineup with an illness. It was the first game he has missed this season. ... The Blues have recalled forward Sammy Blais from the AHL for the fourth time this season. Blais, who leads the San Antonio Rampage with 28 points (11 goals, 17 assists), last played for St. Louis on Dec. 16, when he got hurt against Winnipeg. ... San Jose forward Tomas Hertl missed another game with an upper-body injury.
Sharks: At the Nashville Predators on Thursday.
Blues: Host the Winnipeg Jets on Friday.