It was the longest active drought in the NHL and marked the first time the Sabres won a series-clinching game on the road since the 2007 playoff run, when they defeated the New York Rangers in Game 6 to advance to the Eastern Conference Finals.
They'll await the winner between the Tampa Bay Lightning and Montreal Canadiens, who are heading to a Game 7.
Goaltender Alex Lyon, who took over for Ukko-Pekka Luukkonen in Game 2, has yet to allow more than two goals in a game these playoffs, going 1-1-2-1 in the final four games of the series.
Boston has now been bounced out of the first round three times in the last five years.
Recap & Highlights
The Sabres had the chance to clinch their first-round series win for the first time since 2007, and they made sure to get on the board first.
Alex Tuch scored his fourth goal of the series just 3:25 into Game 6 to give the Sabres an early 1-0 lead.
He became the fourth Sabres player in the past two decades to record three or more road goals in a single playoff series, joining Thomas Vanek (2011), Stu Barnes (1999), and Michal Grosek (1998).
Buffalo wouldn't waste any time doubling their lead.
This time around, it was Mattias Samuelsson who scored his second of the series to make it 2-0 with the Sabres' sixth shot on goal for the game.
To no surprise, the Sabres finished the first period going 0-for-2 on the power play.
It marked the 152nd time the Bruins faced a multi-goal deficit in the playoffs - they only went on to win four times.
But Boston pulled within one just two minutes into the second period.
David Pastrnak scored his third goal of the series to make it 2-1. He joined Brad Marchand (10), Milan Lucic (10), and David Krejci (9) to score eight or more career playoff goals in Game 6 and Game 7 combined.
The second period had a 9:13 stretch without a single whistle, surpassing the seven minutes from the Canadiens-Lightning Game 6 that was going on at the same time.
Buffalo failed on their third power play of the night after Boston's Morgan Geekie recorded the only penalty (interference) of the second period.
And, like the start of the previous two periods, the third would start off with another goal in the first six minutes.
This time, it came from Buffalo's Zach Benson, who scored his second of the playoffs to make it a 3-1 game with 14:02 left.
Both teams had exchanged exactly 21 goals at that point, but it was the Sabres who would go on to prove victorious as they advanced to the second round following a Josh Norris empty-netter to make it 4-1, also his first goal of the series.