Brad Marchand added an empty-net goal with two minutes to play to seal a victory that gave the Bruins a 1-0 lead in the best-of-seven games championship series.
Boston, who had a 10-day layoff after sweeping Carolina in the Eastern Conference final, looked out of sorts early and were staring at a two-goal deficit when Blues forward Vladimir Tarasenko scored one minute into the second period.
The Bruins stormed back 76 seconds later when a cross-ice pass from Kuraly deflected off rookie defenseman Connor Clifton's skate and into the Blues net.
Boston tied the game on the powerplay when Charlie McAvoy fired a wrister from the slot under the glove of Blues goalie Jordan Binnington with seven minutes left in the second.
St. Louis forward Brayden Schenn opened the scoring seven minutes into the game.
The series resumes on Wednesday in Boston.
(Reporting by Frank Pingue in Toronto; Editing by Peter Rutherford)
