The Bruins, outscored 15-9 in the first fives games of the series, were looking for a second straight road win over the Sabres.
Things got off to a rocky start for them though, as Mark Kastelic was sent to the penalty box for roughing just two minutes into the contest. They were made to pay on the ensuing power play, as Rasmus Dahlin took a shot from the Boston face-off zone that went straight in.
Behind once more, there was no further room for error for the guests. They finished the opening frame with nine shots on target in comparison to Buffalo's five.
But, the Bruins did respond midway through the second. After going for goal from a tight angle, Elias Lindholm's initial effort was kept out by Sabres goaltender Alex Lyon. The puck popped into the air, and the Swede was first to it, swiveling and firing it into the net to tie the game.
There was a hush from the home crowd as the Bruins celebrated a much-needed leveller.
The rest of regulation saw the score remain 1-1, with both teams trading blows and trying to find the go-ahead marker. For the first time in the series, overtime was required.
Boston prevailed in both games that went beyond regulation time against Buffalo during the regular season, and kept the streak going when David Pastrnak finished off a stretch pass from Lindholm 9:14 into the first period of overtime to seal the win.
The teams will return to the ice on Friday in Boston as the Bruins look to force a deciding Game 7.
Wild beat Stars to close in on series win
In Dallas, the Stars fell 4-2 to the visiting Minnesota Wild as the guests have now taken a 3-2 series lead.
It did not take long for the opening goal to come about, as Mats Zuccarello received a pass right in front of the net from Kirill Kaprizov and converted to give the Wild the lead.
The Stars did not panic though, tying up the game five minutes later on the power play. Miro Heiskanen took a shot from just inside the blue line past Jasper Wallstedt to make it 1-1.
Matthew Boldy thought he had put the Wild back in front just before intermission, but a successful coach challenge left the score level for the break. However, the forward would not be denied 32 seconds before the end of the second. Left with a bit too much space, he shot from point-blank range to put the guests back ahead.
It became 3-1 in the third when Mike McCaron was the fastest to the puck, breaking in on goal and beating Jake Oettinger to double his side's advantage.
Dallas, having lost the last two games, were in danger of falling behind in the series for the first time. They did find a goal with 3:21 left to restore some hope. Jason Robertson made the most of the opportunity to make it an exciting finish.
Ultimately, that seldom mattered, as Kaprizov scored an empty netter shortly thereafter to round out the scoring and give Minnesota the chance to win the series on home ice.
Game 6 is on Thursday at the Grand Casino Arena, with the Colorado Avalanche awaiting the winner of the series.