Tage Thompson scored the only goal in the shootout after Buffalo tied it 2-2 when Victor Olofsson scored from the top of the slot with 3:41 left in the third period.
The Sabres lost 3-2 to the Rangers in overtime Thursday to open the two-game set here.
"We were disappointed with some things the other day," Buffalo coach Don Granato said. "Our internal demand, our personal standard, we didn't hit the other day for whatever reason. We spoke of it after the game, and I believe the guys took that to heart and elevated and fought through a lot in this game. There were emotions and things that could be very emotional within the context of the game, and we battled right through it, stayed right on it, and we were rewarded for it."
Casey Mittelstadt scored for the Sabres (8-23-6), who have won two of three games since losing 18 in a row, the longest stretch without a victory since the shootout was implemented in 2005-06, eliminating ties. Buffalo is 2-0-2 in its past four.
"Even through the losing streak it was so tough, but the boys were sticking together and working hard and trying to play together," Mittelstadt said. "And then obviously the last few games it's all come together and we're playing a lot better. You can feel it on the bench, just the belief and the battle out of everyone. Just the mood in the locker room has been great, so something to build on definitely. We've still got a long way to go."