Goaltender Alex Lyon was 40 seconds away from his first career playoff shutout before Boston's Sean Kuraly scored his team's only goal of the game at the end of the third period.
Lyon has now allowed only two goals on 55 shots these playoffs after taking over for Ukko-Pekka Luukkonen in Game 2.
Boston will look to avoid elimination in the first round for the third time in the past five years.
This series heads back to Buffalo for Game 5 on April 28th.
Recap & Highlights
The Sabres came out of the gate unloading on the Bruins, putting up four goals on 15 shots, before finishing the period with 19 total, all while holding Boston to six shots.
Buffalo's first goal of the game was scored by Peyton Krebs, his second of the playoffs, with an assist by Alex Tuch, his third, to make it 1-0 just 4:15 into the first.
Krebs and Owen Power recorded a point in each of their first playoff games, joining Danny Gare (1975) as the only players in Buffalo Sabres history to do so.
Boston would receive the first penalty of the game, less than a minute after Krebs' goal for too many men on the ice, but Buffalo would do nothing with their first power play.
However, they would find the net for a second time shortly after it ended as Josh Doan scored on a tip-in for his first goal of the playoffs to make it 2-0.
It would take the Sabres two minutes to find the net for a third time.
After scoring his first goal of the playoffs, Doan tallied his first assist on Zach Benson's first goal as the Sabres took a 3-0 lead before the halfway mark in the first.
The Sabres put up three goals in the opening period of a playoff game for the first time since they did it in 2011 against the Flyers in Game 6.
Bowen Byram scored the fourth goal for the Sabres, and his third of the playoffs, to go up 4-0 after Bruins goaltender Jeremy Swayman had his pad tangled up with his own defenseman in front of the net.
Byram became the third defenseman in franchise history to have a goal in three consecutive playoff games, joining Mike Ramsey (1983) and Ken Sutton (1993).
The last time the Bruins allowed four goals in the first period of a playoff game was in 1991 against the Hartford Whalers.
Despite both teams finishing the first period with 12 hits each, Boston's 10 giveaways, multiple in their own end, to Buffalo's five proved to be the big game changer.
Boston played noticeably better in the second period, but neither team got anything going as they both failed on each of their lone power play opportunities.
Five minutes into the third, Buffalo began to increase their lead even more, all in the span of a minute.
Beck Malenstyn scored his first career playoff goal in his eighth playoff game to make it a 5-0 game.
Then 1:24 later, Tuch would join in on the scoring with his third goal of the series and his sixth point to make it 6-0.
His goal marked nine consecutive for the Sabres in the series as Boston finally made the move to take Swayman out for the game.
Buffalo would go on a five-minute power play to finish off the game when Boston's Kuraly scored the team's lone goal with 40 seconds left to go, ending the Sabres' consecuitve goal streak.