"You got to take them when you can," Eichel said.
Sabres goalie Robin Lehner made 12 of his 33 saves in the third period.
"You go back to all the games he's played recently on home ice, he's given us a chance to win every time he's getting in between the pipes, and we finally got some goal support for him, because they did get some good looks in the second period and [he] made some big saves for us," Buffalo coach Phil Housley said.
O'Reilly gave the Sabres a 1-0 lead at 3:59 of the third period when he took a shot from the side of the net that snuck through Elliott's pads. Kane made it 2-0 at 11:08 when he deflected a shot from Josh Gorges, who returned to the lineup after he was a healthy scratch the past two games.
"We came out really good in the second," Lehner said. "[The] other team's going to have some momentum in the game, but I think we took it back. I think we did the right things and we made them earn their opportunities. ... That's a good step forward."