The Hurricanes lead the series with Game 4 here Sunday.
"You do everything you can to win. … It was a great team effort," Swayman said. "My first playoff start] was special. … It's a great experience every time you walk into a full building here, and playoffs are a whole different level. It was great to get the win in front of our hometown tonight."
Brad Marchand had a goal and two assists, and David Pastrnak and Charlie Coyle each had a goal and an assist for Boston, the first wild card in the East. The Bruins were 5-for-5 on the penalty kill.
"The penalty kill was good for us tonight, obviously," Boston coach Bruce Cassidy said. "It has been the whole series. Our power play caught up to it tonight, so that helps. Hopefully, we're a little cleaner out of the gate on Sunday, but playing with the lead, we've talked about it enough. It makes it different for both teams, and it certainly helped us tonight."
[RELATED: [Complete Hurricanes vs. Bruins series coverage]
Vincent Trocheck and Jaccob Slavin scored for Carolina, the No. 1 seed in the Metropolitan Division. Pyotr Kochetkov made 24 saves in his first playoff start. The rookie made 30 saves on 32 shots in relief of the injured Antti Raanta in Game 2, a 5-2 win Wednesday in his playoff debut.
"[Kochetkov] was fine. You can't blame him on any of the goals," Hurricanes coach Rod Brind'Amour said. "They were all back-door tap-ins and great players ripping good shots in pretty good areas. The top guys have to be better. It's simple."
Linus Ullmark started the first two games of the series for Boston, allowing four goals on 24 shots during a 5-1 loss in Game 1 on Monday before giving up four goals on 33 shots during a 5-2 loss in Game 2 on Wednesday.
Trocheck put the Hurricanes ahead 1-0 at 9:17 of the first period when he batted in the rebound of Brendan Smith's point shot.
Coyle tied it 1-1 with a shorthanded goal at 17:16, redirecting Jake DeBrusk's pass on the rush.
"We know what the issue is," Brind'Amour said. "We had a decent first period. Then the power play gave up a shorty, and they got some life out of it. If you give up too many looks to their top guys, they're going to make you pay. That's what happened."