Paul Cotter and Timo Meier scored, and Jake Allen made 17 saves for the Devils (2-0-0), who won 4-1 here on Friday.
"It's big," Cotter said. "It's the start of the season and toward the end of the year, every team knows it could be a matter of two points, one point, so these are the ones that count. Obviously, what an experience for us but it's always awesome to go home with two wins."
Tage Thompson scored for the Sabres (0-2-0), and Devon Levi made 34 saves in his season debut.
"You've got to give them credit, they played hard," Thompson said. "They didn't make it easy on us. We just got to find a way to bear down and get another one. For the most part, one goal is not going to win you a game. We know that and we've got to find a way to score more."
Thompson gave Buffalo a 1-0 lead at 8:18 of the second period when he stuffed a loose puck under the pad of Allen at the right post.
"I thought our first 40 minutes were really strong," Thompson said. "There's one or two mistakes we make in the third and they capitalize on it."
Casey, playing his second NHL game, tied it 1-1 with a power-play goal at 11:59. After taking a pass at the left point, the 20-year-old defenseman took a few strides and sent a wrist shot into the top left corner.
"This whole week has been a blast," Casey said. "The guys have just been great, welcoming me in. I talked about it before ... it's not your normal first experience in the NHL, but it's been super fun and to get two wins has just been awesome. It was an awesome way to start right."