Sean Kuraly scored the game-winner at 5:21 of the third period, and Connor Clifton, Charlie McAvoy and Brad Marchand scored for the Bruins, who fell behind 2-0 on Vladimir Tarasenko's goal at 1:00 of the second period. Tuukka Rask made 18 saves.
Brayden Schenn had a goal and an assist, and Jordan Binnington made 34 saves for St. Louis.
[WATCH: Blues vs. Bruins Game 1 highlights | Complete series coverage]
Game 2 is at TD Garden on Wednesday (8 p.m. ET; NBCSN, CBC, SN, TVAS).
"We weren't too happy with the way the game was going," Bruins coach Bruce Cassidy said. "We weren't playing our game, and I think after the [Blues'] second goal, our guys, that kind of woke us up. We needed something good to happen, and it did."
Clifton kept his stick on the ice and redirected Kuraly's cross-slot pass to make it 2-1 at 2:16 of the second. The goal came 1:16 after Tarasenko extended his point streak to seven games (nine points; four goals, five assists) with a goal off a Boston turnover.
"It was a good changing point," Clifton said.
The Bruins tied it 2-2 on McAvoy's power-play goal at 12:41 when his shot deflected off Blues defenseman Alex Pietrangelo's stick.
Video: STL@BOS, Gm1: McAvoy rips shot by Binnington for PPG
Kuraly, who had an assist, made it 3-2 after Binnington couldn't handle a wrist shot by Bruins defenseman Zdeno Chara. The puck dropped in the slot, and Boston forward Noel Acciari found it and fed Kuraly by the right post. He played the puck off his skate to his stick and slid it past Binnington.
"I've got to be better," Binnington said. "That's my puck there."
Marchand scored into an empty net with 1:49 remaining for the 4-2 final.
Boston outshot St. Louis 18-3 in the second period, 30-12 in the final two periods and 38-20 in the game.
"I think our pace was really set by our defensemen," Bruins defenseman Torey Krug said. "It was sluggish in the first. The second period, we were able to just get around pucks and move it north very quickly, and then our forwards weren't trying to do any east-to-west plays. They were just chipping it behind their D and going to work. When our pace is like that, it's tough for a team to hang with us."
Schenn gave the Blues a 1-0 lead at 7:23 of the first period. Their response after Clifton's goal wasn't good.
Video: STL@BOS, Gm1: Schenn nets first goal of Cup Final
"Turnovers," St. Louis coach Craig Berube said. "The second period, I didn't think we skated very well. Also, penalties were a part of it all, but the second period we got pucks and we didn't skate, didn't advance them, turned them over, gave them momentum."
The Blues put the Bruins on the power play twice in the last 6:45 of the first period and twice more in the first 11:04 of the second. They finally connected on McAvoy's goal, but killing that many penalties affected the Blues' flow and their ability to roll four lines.
Boston, which entered leading the Stanley Cup Playoffs on the power play (34.0 percent), was 1-for-5 on the man-advantage.
"It takes a lot of guys out of the game and that burns up a lot of energy from other guys that are killing all the time," Berube said. "It's too much. We've got to be better there. We've got to be more disciplined."
Video: Clutch Performance: Kuraly scores the game-winner
The Bruins also started managing the puck better after Clifton's goal, which led to cleaner breakouts and easier zone entries. They took the Blues forecheck out of the game.
"We got back to playing our game in the right way," Bruins center Patrice Bergeron said.
They said it
"Come the second period, we picked up our game. We got more physical and we started moving our feet, moving the puck and we ended up spending more time in their end and didn't give them as much room. That's when we started taking the game over." -- Bruins forward Marcus Johansson
"When that first goal went in, they got some momentum and they had a big push there. We held it to a tie game going into the third, and it wasn't the outcome we wanted. But it was good to feel it out, and we'll be back at 'em for Game 2." -- Blues goalie Jordan Binnington
"We just know how to check. We've done it to (the) Carolina [Hurricanes], Toronto [Maple Leafs], Columbus [Blue Jackets] (in the first three rounds). We know how to play winning hockey when we need to." -- Bruins coach Bruce Cassidy
Video: Dunkin' NHL Comebacks: Bruins rally back in Game 1
Need to know
Since 1939, when the NHL started using the best-of-7 format in the Stanley Cup Final, Game 1 winners have won the Cup 77.2 percent of the time (61 of 79 series). Teams that have won Game 1 in any best-of-7 series have a series record of 476-219 (.685). Game 1 winners this season are 9-5 (.643). … The Bruins have won eight straight games since losing Game 3 of the Eastern Conference Second Round against the Blue Jackets, outscoring opponents 32-11. … Defenseman Matt Grzelcyk was the only one of Boston's 18 skaters without a shot. … The Blues are 0-13 in Cup Final games; they last reached the Final in 1970, when they lost to the Bruins. … St. Louis was playing its first game in six days.
Game 2 of Stanley Cup Final at TD Garden on Wednesday (8 p.m. ET; NBCSN, CBC, SN, TVAS)
Video: Bruins rally past Blues in Game 1