Boqvist gave the Bruins a 1-0 lead at 4:01 of the first period, scoring in front off Anthony Richard's centering pass.
"It kind of just happened," Boqvist said. "I don't think it's going in if I just one-touch it, so it kind of just happened. It was nice to see that one go in, and first goal here at home too."
Johnston tied it 1-1 at 11:14 when he redirected Thomas Harley's shot. Johnston has six points (three goals, three assists) during a three-game point streak.
"I think that's one of the things that sticks out here is everyone kind of stays level," Dallas defenseman Alexander Petrovic said. "No one's freaking out or going nuts. It's a hockey game. Things happen quick."
Suter gave the Stars a 2-1 lead at 1:04 of the second period on a shot from the left face-off circle.
The Bruins tied it 2-2 28 seconds later, when Brazeau scored from in front at 1:32. It was the forward's first NHL goal in his first game. He had signed a two-year, two-way contract earlier Monday.
"It's a cool feeling," Brazeau said. "It's been a long time coming for me. I've worked hard to get here, so to see that kind of comes full circle. It feels really good."