Pavel Zacha scored his first NHL playoff goal in his 25th game, and Swayman made 26 saves for Boston, the No. 2 seed from the Atlantic.
"I'm just overwhelmed with emotions," Swayman said. "It's not about me as an individual, it's about our team, and to see it all end so abruptly is something you never want to feel, and I'm just so proud of everyone and being here and setting the foundation for what's to come next."
Zacha gave the Bruins a 1-0 lead at 19:07 of the first period on a partial breakaway, scoring with a backhand off a pass from Jake DeBrusk.
The Bruins were held to two goals or fewer in each of final five games of the series, scoring a total of eight goals since a 5-1 win in Game 1.
"It was an honor to coach them because of the hard work and the competitiveness that they brought throughout the year," Bruins coach Jim Montgomery said. "That was the biggest thing I wanted to get across, and that you're never happy when you don't win your last game of the year. And some players had tremendous growth this year, and they can be proud of that and in the end, we have to find ways to get better and win our last game of the year."