Martin Jones made 28 saves for San Jose, which improved to 7-2 at home in the Stanley Cup Playoffs.
"It was big that we got the Game 1 win," said Sharks captain Joe Pavelski, who scored in the second period. "That was big. We've had different guys step up along the way. Tonight, forwards scored some goals, D-men were great breaking pucks out at times. We'll expect [the Blues] to be better."
[WATCH: Blues vs. Sharks Game 1 highlights | Complete series coverage]
Game 2 of the best-of-7 series is at San Jose on Monday.
Joel Edmundson, Ryan O'Reilly and Tyler Bozak scored, and Jordan Binnington allowed five goals on 24 shots for St. Louis, which had 12 giveaways in its second loss in seven road playoff games.
"It wasn't our best game," O'Reilly said. "We didn't take care of the puck well enough. They outplayed us tonight. ... It's disappointing, but we have to respond. It's a long series, and we know we have to be better next game."
Couture gave the Sharks a 1-0 lead at 3:31 of the first period when he scored with a one-timer on a 2-on-1 with Nyquist.
Video: STL@SJS, Gm1: Couture buries Nyquist's feed
Edmundson tied the game 1-1 at 9:13 when he tipped a shot by Jaden Schwartz for his first goal of the playoffs.
Pavelski gave the Sharks a 2-1 lead at 11:24 of the second with a 5-on-3 power-play goal from the side of the net. Binnington stopped the initial shot with his right pad, but Pavelski batted in the rebound.
Kevin Labanc made it 3-1 at 7:41 of the second with a wrist shot for his first goal in seven games. Joe Thornton tried to shoot the puck into the zone, but it bounced off Labanc, who got to it just inside the blue line, cut across the left face-off circle and scored with a wrist shot from the slot.
"The playmaker he is ... he tried to shoot it in, but I mean it's hockey, it's a fast game, and the puck popped right out to me," Labanc said.
Video: STL@SJS, Gm1: Labanc rifles wrist shot by Jones
O'Reilly got the rebound of a David Perron shot to the right of the net, pulled the puck around Jones' pad, and tucked it inside the right post to make it 3-2 at 8:58.
Meier, who had three goals in San Jose's first 14 playoff games, made it 4-2 at 10:24. After Couture forced Colton Parayko into a turnover at the St. Louis blue line, Meier got the puck, skated around defenseman Jay Bouwmeester, deked, and scored with a backhand.
"Great play by Logan," Meier said. "That's an unbelievable stick there to get the puck away from the defenseman. That goal wouldn't happen if that play wasn't made by Logan."
Meier scored again at 17:34 to make it 5-2. His shot from below the goal line went in off defenseman Vince Dunn's skate.
Video: STL@SJS, Gm1: Meier tallies off defender for second
Bozak got the Blues to within 5-3 at 13:01 of the third.
St. Louis outshot San Jose 12-5 in the third period.
"We're not worried," Blues forward Robert Thomas said. "We've had our backs against the wall the entire second half of the year. We know we'll respond. I was happy with our third period, but it was a little late. We'll bring that into Game 2."
Couture scored into an empty net with 2:21 remaining for the 6-3 final. It was his NHL-leading 11th goal.
"We were too spread out," Blues coach Craig Berube said. "In playoff time, you've got to have numbers around the puck and have puck support everywhere. We didn't have enough support."
Video: STL@SJS, Gm1: Berube on Game 1 loss to Sharks
The line of Meier, Couture and Gustav Nyqvist combined for eight points (four goals, four assists).
"I thought Timo and Gus played really well," Couture said. "[Meier] was a bull. When he skates and (with) his strength, he's tough to defend. We're fortunate enough to play with him on our line."
The Sharks won Game 1 for the third straight series.
"You need complete games in the playoffs right now, and tonight was a good step in the right direction," Pavelski said.
Video: Clutch Performance: Couture leads Sharks in Game 1
They said it
"There were a few too many mistakes out there. Whenever there was, they capitalized on every chance they got. ... Overall, they were just a better team. We'll be ready for Monday." -- Blues defenseman Joel Edmundson
"I thought that third period, we got to our game, we were aggressive, we're on the forecheck, controlled the puck in the offensive zone. We didn't get there enough tonight." -- Blues coach Craig Berube
"You don't care how this time of year, you just want to get the win. It was important to get off to a good start. I think both teams looked a little like they were coming out of Game 7s, and it was a little sloppy in areas, both ways, but we found a way." -- Sharks coach Peter DeBoer
Need to know
Erik Karlsson, Brent Burns and Marc-Edouard Vlasic each had an assist for the Sharks, who lead the NHL in points by defensemen with 38 (eight goals, 30 assists). The Blues are second with 33 (five goals, 28 assists). ... Bouwmeester's interference penalty at 9:36 of the first period was the Blues' first penalty in 136:19 dating to Game 6 of the second round against the Dallas Stars. ... The six goals allowed by the Blues tied their playoff high (6-3 loss to Winnipeg Jets in Game 3 of first round on April 14).
Game 2 of Western Conference Final at SAP Center on Monday (9 p.m. ET; NBCSN, CBC, SN, TVAS)
Video: Couture, Meier power Sharks past Blues in Game 1