Tyler Toffoli scored twice, and Alexander Holtz, Luke Hughes, Dawson Mercer and Palat each had a goal and an assist for the Devils (8-9-1), who ended a three-game losing streak. Vitek Vanecek made 10 saves, and Bratt and Dougie Hamilton each had two assists.
New Jersey was coming off a 2-1 home loss to the Columbus Blue Jackets on Friday.
"There was no real panic within our game and we were calm and we made all those little plays that are not easy to make," Toffoli said. "You don't always want to make them; you want to kind of make the home-run plays all the time, but I thought we made those little plays and it went a long way for us."
Jeff Skinner and Kyle Okposo scored for the Sabres (9-10-2), who defeated the Pittsburgh Penguins 3-2 at home Friday. Eric Comrie made 14 saves on 18 shots and was replaced to start the second period by Devon Levi, who made 17 saves.
"It's very disappointing; this was self-inflicted, and we just didn't work hard enough," Buffalo coach Don Granato said. "We had a struggling first period. There were too many goals on breakdowns, and that's how goals are scored in the NHL."
New Jersey scored four goals in the first to take a lead into the second on home ice for the first time in 10 home games this season (4-5-1).
"It was one of those games we've been searching for, and definitely the past couple games have been really disappointing within our group and it's been really frustrating," Toffoli said. "So we've got to use this as a stepping stone and know that's the bar for us."
Holtz made it 1-0 from the slot at 1:31, and Toffoli pushed the lead to 2-0 at 11:09 on a backhand just outside the left post.
Hischier extended it to 3-0 at 15:20 when the puck deflected in off his leg while he skated down the slot, before Palat made it 4-0 on the rush at 18:27 with a snap shot in the high slot.
"We got to play as a team and I think tonight we did that," Hischier said. "It was a 60-minute effort. We're a dangerous team, but I think we understand that. We’ve just got to execute."