Sam Reinhart had two assists for Buffalo (10-6-3), which was 0-4-2 in its previous six games. Linus Ullmark made 21 saves.
"I just think for us to win, we have to play a little bit cleaner of a game, but it's good to get out of a little hole that we were in there and find a way to get two points," Eichel said.
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The Sabres were without injured defenseman Zach Bogosian (hip), and forwards Johan Larsson (upper body), Vladimir Sobotka (lower body) and Marcus Johansson (upper body).
"Tonight was going to be a grind, we knew that," Buffalo coach Ralph Krueger said. "We definitely, with the injuries, needed our best players to step up, and Jack showed a world-class performance today and just excellent leadership that I also told you he's been showing, as I've told you, in the adversity of the last few weeks. It's really probably the fruit of all that attitude that he's had, [it] came out in the goals tonight."