Jimmy Vesey and Victor Olofsson scored, and Sam Reinhart had two assists for the Sabres (16-11-6), who have won three in a row and are 5-1-3 in their past nine games. Linus Ullmark made 36 saves.
"Obviously, when you get on a run like this and the team's playing well, you're just really trying to do the little things every night. It's rewarding," Eichel said. "You try and take care of your end and try and be responsible and not do too much, I think good things happen. I've been fortunate. I've gotten some good bounces, I've had some good breaks lately, and our team's been playing really well, so you put those things together, I'm the beneficiary of a lot of really good hockey plays out there and just trying to keep it going. Obviously, you find confidence when you go through a streak like this and it's no different for me."
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Matt Duchene, Colton Sissons and Ryan Johansen scored, and Juuse Saros made 28 saves for the Predators (14-11-5).
"We've definitely been taking strides to be a consistent winning club again, so taking a step back tonight is frustrating," Johansen said.
Duchene scored 57 seconds into the first period to give Nashville a 1-0 lead, deking Ullmark before going around his left pad and tucking the puck inside the right post.