It was Swayman's first start since allowing seven goals in a 7-2 loss at the Ottawa Senators on Monday.
"Really good," Bruins coach Marco Sturm said of Swayman. "It doesn't really surprise me. I expected him to be just like he was today -- outstanding."
Casey Mittelstadt had a goal and assist, and Viktor Arvidsson had a goal for the Bruins (7-7-0), who have won three in a row and four of five. Charlie McAvoy had two assists.
"It was important for us," Swayman said. "We wanted to end this homestand the right way, and we played a complete game today, which is something we can really build on. It's good to have momentum with three (wins) now and feeling good about our game and moving forward."
Alexander Nikishin scored, and Frederik Andersen made 20 saves for the Hurricanes (7-4-0), who have lost three of four.
"We actually got better as the game went on," Hurricanes coach Rod Brind'Amour said. "The first period, they only had 11 shots on net, and five in the third, so we did a lot of good things. The problem is the two (goals) that we gave up, our 'D' got a little too aggressive, got caught and it cost us."
Mittelstadt gave Boston a 1-0 lead at 1:27 of the third period. McAvoy jumped into the play, received a pass from Pavel Zacha, kicked it from his skate to his stick and fed Mittelstadt, who scored from in front for his fourth goal of the season.
"It was nice to get one early in the third and play with the lead," Mittelstadt said.