Tarasenko put St. Louis in front 3-2 at 18:12 with a turnaround shot after he received a backhand, between-the-legs pass from Robert Thomas on a 3-on-2.
"[Tarasenko's] line had some looks," Blue coach Craig Berube said. "He's playing a solid, all-around game. Got him out there [at the end with Boston's] goalie out, killing it off. He's doing a good job all-around for us, and leadership-wise, too."
Tarasenko extended the lead to 4-2 at 9:27 of the third with a one-timer on a 3-on-1 after the Bruins turned it over in the offensive zone.
"It's just really not very intelligent hockey, and you don't win against good teams when you don't play intelligent hockey," Cassidy said. "I think our effort -- we're trying and guys are working hard, we've just got to be smarter, plain and simple."
Bruins defenseman Brandon Carlo left the game with an undisclosed injury at 10:06 of the first. Cassidy said Carlo was "not feeling well."
NOTES: Thomas extended his point streak to 10 games (six goals, 12 assists). … Tarasenko's multigoal game was his 37th in the NHL, passing Brendan Shanahan for sole possession of fifth in Blues history. … Bergeron scored his ninth career opening-minute goal, tying Ray Bourque for the most in Bruins history. The only active players with more are Sidney Crosby of the Pittsburgh Penguins (13), Joe Thornton of the Florida Panthers (11), Alex Ovechkin of the Washington Capitals (10) and Anze Kopitar of the Los Angeles Kings (10).