Sean Monahan had a goal and an assist, Johnny Gaudreau and Tobias Rieder scored, and Derek Ryan had two assists for the Flames (14-12-4), who have won three in a row and are 4-0-1 in their past five games after going six straight without a win (0-5-1). David Rittich made 26 saves.
"We're trending in the right direction, and our game keeps building the way we want it to go," Lucic said. "I think not only that, we're getting contributions throughout the whole lineup. Everyone's a part of it, and we're going to need more of that moving forward."
WATCH: [All Sabres vs. Flames highlights]
Sam Reinhart, Jake McCabe and Jack Eichel scored, and Victor Olofsson had two assists for the Sabres (13-11-5), who are 2-1-2 in their past five. Linus Ullmark made 25 saves.
"I don't think we played a very good game," Olofsson said. "I feel like we didn't play well at all. Started OK, and then Calgary, I thought they took over. They were a lot better than us today up until the last few minutes of the game where we kind of woke up, and we fought to the end but couldn't get that last one in."
Rieder, playing his 400th NHL game, put Calgary up 2-1 at 11:18 of the second period when he slid a backhand between Ullmark's legs on a shorthanded breakaway.
"It always feels good, especially on the PK," Rieder said. "It was good. It was at the right time of the game, as well, it got us going. It was a big goal."