Buffalo had a 2-1 lead but gave up a shorthanded goal early in the third period that cost them a better opportunity to jump New Jersey in the Eastern Conference standings.
With the win, Buffalo sits in 11th place with 38 points, four points behind the seventh-place New York Rangers. New Jersey is in ninth place with 40 points.
The Sabres are winners of their past three games; the Devils are headed in the other direction.
Buffalo also is no stranger to tight games. The Sabres have had 15 of their past 20 decided by one goal and are 7-3-5, overall 10-8-6 in 24 one-goal games this season. Buffalo improved its record in the shootout to 5-4.
"I thought we played pretty solid. We had some ups and downs in the game, obviously. Some young mistakes, that happens," Sabres forward Steve Ott said. "I mean, all of us are guilty of a lot of mistakes out there but [Miller] was phenomenal again for us. When we have a goaltender like that backstopping us, it makes everybody a little bit more confident on that ice."
Miller made 37 saves for Buffalo through 65 minutes. Brodeur turned aside 23 shots.
The Devils, having lost their three previous games in regulation, are 2-5-5 in their past 12, 0-3-4 without injured forward Ilya Kovalchuk (shoulder).
Forward Dainius Zubrus had an assist against the Sabres and knows time is winding down to collect points. New Jersey has nine games remaining.
"We [need to] get two points. Every game is a huge game and it's a race now. It's not a marathon. It is a sprint and you want to get every point," Zubrus said. "So it's frustrating, but if we play the same way, I will not be shocked if we win the rest of the games here."
The Devils tied it 2-2, 6:46 into the third period while shorthanded. Zubrus carried the puck around the Buffalo net without much heat on him and sent it to Fayne at the top of the zone. The defenseman wristed a shot that beat Miller stick-side. The goal was Fayne's first of the season.
Buffalo has allowed a League-worst seven shorthanded goals this season.
"That's a tough one. All five of us are accountable for going hard on that guy, Zubrus, carrying the puck around the net," said Ott, who was on the ice for the goal. "We definitely need to fish him out hard and if not, lay a body on him, and that way he doesn't have a great opportunity to make a great play out and score a big shorthanded goal for them."
The Sabres took a 2-1 lead on a perfect tic-tac-toe play in the Devils zone with 6:16 left in the second period. At the right point, Gerbe fired a pass across ice to Cody Hodgson in the left corner. Hodgson then knocked the puck to Ott in the slot, who settled it and roofed a shot over a diving Brodeur.
Ott has scored in three straight games. He also has five points (four assists) in his past five games against New Jersey.
New Jersey took advantage of a young Buffalo team with a solid forecheck. For long stretches of the game, the Devils were able to pin the Sabres in their zone, outshooting the home team 18-8 at one point.
"I liked our bench, and even in those times where we did have some lulls in our game, I think everybody was still saying the right things and we knew what we had to do," Sabres interim coach Ron Rolston said. "It was just getting out there and a couple of shifts in a row just trying to make a difference, and I thought we did a good job there just trying to get through that."
The Devils have had trouble scoring goals of late. They had four in their four games coming into Sunday, and Sullivan's power-play goal was the team's first in five games.
"It's really concerning that we can't just win. That's the bottom line. We played one of the teams that we should beat today and I thought we deserved to beat them the way we played," Brodeur said.
Jochen Hecht turned an Andy Greene giveaway in the Devils' zone into the Sabres' first goal. On a Buffalo dump in, Brodeur played the puck to Greene in the corner. Greene sent the puck up ice, but Hecht intercepted it above the faceoff dots. Hecht then turned toward the net and threw a pass on goal. Kaleta, moving to the crease, tipped it in for his first of the season.
It was Kaleta's first point since March 7, 2012 and his first goal since Feb. 8, 2012.
"It feels good to be able to contribute offensively. I want to be that person where I'm all around and I worked at it pretty [hard], so to be able to get it in the net is great," Kaleta said. "But ultimately in the game we have to be better. Give us a credit for being resilient and sticking with it, but ultimately we need to tighten up and be better for a full 60 minutes."
Miller made a highlight reel-save with 2:06 to go in the first period when he stopped Sullivan with his paddle to prolong a shift in which the Devils brought about considerable pressure. That 2:20 shift for Buffalo's Luke Adam, Kevin Porter, Brian Flynn, Mark Pysyk and Adam Pardy finally ended when Porter was sent to the box for cross-checking Patrik Elias in the corner.
Sullivan would not be denied in his next opportunity. On the faceoff, Ott won the draw, but Devils forward Steve Bernier picked off defenseman Tyler Myers' soft pass around the end boards. Bernier skated to the front, passed the puck into the crease, and Sullivan redirected it out of the reach of Miller's glove.
Sullivan's goal was his first since being reacquired by the Devils via trade from the Phoenix Coyotes on Wednesday.
He played in a New Jersey uniform for the first time Saturday against the Toronto Maple Leafs since his last stint with the Devils ended Feb. 20, 1997. Sullivan's prior goal with the Devils came on Feb. 15, 1997 against the Montreal Canadiens. Brodeur and Saku Koivu (Anaheim Ducks) are the other players from that game still playing in the NHL.
Sullivan was New Jersey's first shooter in the shootout and his backhand opportunity was stopped by Miller.
"He's got a good shootout record. I thought he did a great job for us tonight," Devils coach Peter DeBoer said. "[The Elias-Travis Zajac-Sullivan line] spent the whole night in the other team's end. They didn't have anything to show for it 5-on-5, but it wasn't for lack of effort."