St. Louis won for the fifth time in seven games and bounced back from a shootout loss to Ottawa on Tuesday.
Boston rallied from a 2-0 deficit to force the extra period, but wound up losing in overtime for the fourth time this season.
Alexander Steen scored in the first period and Jaden Schwartz added a second-period goal to put the Blues in charge heading into the third.
David Krejci and Brad Marchand then scored in a span of 2:08 of the third, allowing the Bruins to pick up a point. Boston's lead in the Atlantic Division grew to seven points over Tampa Bay.
Boston captain Zedno Chara missed the game, having gone to the Olympics early in order that he could carry the flag for Slovakia in Friday's opening ceremonies.