Haula snapped a tie 14:17 into the third period, putting New Jersey ahead to stay. He also scored a short-handed goal in the second.
"I think we could have maybe played a little bit better and but it's early and we're all trying to get better," Haula said. "We're just happy we got two points."
Haula missed out on a hat trick when his stick snapped as he was about to shoot toward an empty net with just over a minute remaining.
"It's a little sad. I don't have a hat trick, so I don't how much closer to that I'm going to get," he said. "But it would have been worse if they tied that up."
Buffalo had won two of three, including a 6-4 victory at Ottawa on Tuesday. Tage Thompson, Rasmus Dahlin, Dylan Cozens and JJ Peterka scored for the Sabres, who dropped to 3-6-2 in their last 11 games against Devils.
Eric Comrie made 16 saves for Buffalo before departing in the second period with a lower-body injury. Ukko-Pekka Luukkonen came in and finished with 19 stops. Devon Levi, another Buffalo goaltender, also is out with a lower-body injury.
"We play 82 games. It's a grind and this is evidence of it," Sabres coach Don Granato said. "We have to deal with it and make the best of it. Ukko is the least of my concerns. He was good."
The 24-year-old Luukkonen, a second-round draft pick by Buffalo in 2017, is in his fourth season with the Sabres.
"I have to be ready. I'm always ready on the bench, talking to the guys. That helps me," the Finnish netminder said. "In the end, it's hockey."
New Jersey grabbed a 4-3 lead when Jack Hughes scored a power-play goal 2:09 into the third. Jack Hughes has five goals and 13 assists in seven games this season.
Cozens tied it at 4 with his third goal at 12:35. Cozens has a six-game point streak.
Haula scored his first goal of the night 13:47 into the second. He sped in on Luukkonen for a breakaway and flipped a backhand into the net.
Dahlin then tied it at 3 at 17:32, scoring his first goal of the year. Dahlin has a seven-game point streak.
Each team scored twice during an entertaining first period.
Peterka rifled a high wrist shot past Vanecek at 6:11 for his third goal this season. Alexander Holtz tied it for New Jersey at 7:41.
Thompson then put the Sabres ahead at 15:42, tapping the puck past Vanecek in a goalmouth scrum for his fourth goal.
"There were pockets of real good and pockets of inconsistency," Granato added. "We need more of the real good."
Bratt tied it again at 16:11, stealing the puck from a Sabres defender and speeding in to beat Comrie for his fourth goal. Bratt is second on the Devils with 11 points.
"We scored a lot of goals so that's always a good thing," said Vanecek, who won for the third time in five games. "Some details can be better but we won the game and that's the most important thing."
Sabres defenseman Connor Clifton was ejected after a high hit on Devils center Nico Hischier along the boards at the 16-minute mark of the first. The Devils captain played in the second but did not return for the third period.
Devils: Host the Minnesota Wild on Sunday.
Sabres: Host the Colorado Avalanche on Sunday.
AP NHL: https://apnews.com/hub/NHL