Gustav Nyquist and Justin Danforth each had a goal, and Elvis Merzlikins made 37 saves for Columbus (32-32-5), which has lost six straight (0-4-2).
"We were in this game. A 2-2 hockey game, and they score. We were doing a good job on the [penalty kill], then one shot and one tip," Blue Jackets assistant Pascal Vincent said. "I'm disappointed for the players. They were doing a great job, and one shot, one tip. But that's what good teams do, and that's why those guys are competing for a Stanley Cup."
Haula put the Bruins ahead 1-0 at 15:09 of the first period, scoring on a loose puck in front of the crease after Merzlikins saved Charlie McAvoy's backhand.
Nyquist tied it 1-1 at 19:32 on a rebound at the top of the crease after Swayman could not control Cole Sillinger's deflection.
"It's just not good enough, and that's something we talked about before the game, that we didn't like our start last game," Nyquist said. "Fortunately, we got in after that period 1-1, and I thought we got our legs under us in the second. It was a tight game."