St. Louis got game-tying goals from Robert Bortuzzo (at 9:37) and Vladimir Tarasenko (at 14:55) in the first period, each after a Bruins goal by Charlie Coyle (4:44) and Joakim Nordstrom (10:17).
Jordan Binnington made 21 saves, including 18 straight after allowing Coyle's power-play goal on the first shot he faced, and to Nordstrom on Boston's fifth shot. Binnington is 6-2 with a .935 save percentage in games after a loss in the playoffs.
"The boys played a great game, a disciplined game and played as a team for 60 minutes, or more than 60 minutes, I guess," Binnington said. "I think we deserved that one."
Boston, which has scored at least once with the man-advantage in six straight games, didn't score on its next four power plays after Coyle's goal, including one it received with 6:38 remaining in the third period when St. Louis center Brayden Schenn took a slashing penalty.
"It's on us to execute better at the end of the day and move a little bit more," Bruins coach Bruce Cassidy said. "I thought our whole game was like that. We didn't have as much energy it seemed with the puck to create offense 5-on-5 and on the power play."