Game 3 is at Carolina on Tuesday.
Grzelcyk gave Boston a 1-0 lead at 15:22 of the first period when his shot from the left face-off circle trickled under Mrazek's arm.
"I wasn't happy about the first goal," Mrazek said. "Obviously, I have to have that. I knew it went through me."
Jake DeBrusk scored a power-play goal to make it 2-0 at 18:32 after getting his third rebound at the top of the crease.
"Yeah, we got a little frustrated after the second one went in," Hurricanes coach Rod Brind'Amour said. "But again, we weren't very good after the first period or the last five minutes of the first and throughout the rest of the game. We got off our game, and give them credit, there's a reason we got off our game. They're playing their game, and we haven't gotten to ours."
Connor Clifton scored his first NHL goal at 3:46 of the second period to extend the lead to 3-0.
"It was great, something special," Clifton said. "Something I will always remember. It was a solid effort by everyone, and it was a great win."
Grzelcyk made it 4-0 at 17:56 on a backhand shot from the right hash marks on the power play. It was his first multigoal game in the NHL.
David Backes made it 5-0 1:10 into the third period, and Heinen scored off a pass from Patrice Bergeron on a 2-on-1 to make it 6-0 at 4:32.
Williams scored for Carolina at 11:17 on a redirection to make it 6-1, and Teravainen scored into an open net at 17:32 after intercepting a clearing attempt from Rask, who came out to play the puck.
"It's 2-0 and we have been in this situation before (against the Washington Capitals in the first round)," Mrazek said. "We are going home, and we know that we can do it at home and we have to shake it off and think about the next one."
