Rasmus Dahlin tied it 3-3 at 17:32 with his first goal of the season, on a slap shot from the right point.
"Overall, there were pockets of real good and pockets of inconsistency, so we need more of the real good," Buffalo coach Don Granato said. "We did it to ourselves. We made our own mess. It's disappointing."
Jack Hughes’ power-play goal gave the Devils a 4-3 lead at 2:09 of the third, but Cozens tied it 4-4 at 12:35 with a sharp-angled shot inside the right post.
"It's early and we've got a ways to go, and there's a lot of areas we still can get better at," Haula said. "I think everyone would agree with that as individuals and as a team. So, there's lots of hockey left and we just keep building."
Haula missed an opportunity for his first NHL hat trick when his stick snapped in half prior to a shot attempt into an empty net from the right circle with 1:01 remaining in the third.
"It's a little sad," Haula said. "I don't know how much closer to that I'm going to get."
Sabres defenseman Connor Clifton was given a match penalty after he delivered a high hit to Devils center Nico Hischier along the boards at 16:00 of the first. Hischier played in the second period but did not play in the third.
Clifton will have a hearing with the NHL Department of Player Safety on Saturday.
"He was kept out of the third for precautionary reasons," Ruff said of Hischier. "We'll see how he's doing tomorrow."
NOTES: Granato had no update on Comrie after the game. ... Dahlin extended his point streak to seven games (eight points; one goal, seven assists), the longest of his NHL career. ... Cozens ran his point streak to six games (seven points; three goals, four assists). ... Jack Hughes extended his season-opening point streak to seven games (18 points; five goals, 13 assists), the third-longest such run in franchise history, behind Bratt (11 games, 2022-23) and Tim Higgins (10 games, 1984-85). … All four wins for the Devils this season have been comeback victories.