Vanek led the way with his NHL-leading 23rd and 24th goals.
"When they scored two, it crushed a little bit of our energy," Vanek said. "We slowly worked our way back into it, and dominated the last 20 minutes."
Derek Roy scored the go-ahead goal midway through the third period, and Drew Stafford added a goal in the Sabres' fourth win in their last five games.
Patrick Lalime, making his first appearance in seven games, made 20 saves. Lalime improved to 2-4-1, ending a personal five-game winless streak.
"From the first to last second, I think we played a really good road game," Lalime said. "We didn't give them much, especially in the second period when we turned it around."
Patrik Elias and Bobby Holik scored for New Jersey. The Devils had won four in a row and nine of their previous 10 games.
"We really didn't control the puck tonight," Elias said. "For the most part we played into their hands. Instead of calming down and controlling the game, we were a little hasty."
Vanek got Buffalo rolling 1:10 into the game, steering in the rebound of Jochen Hecht's backhander off Kevin Weekes' pads.
"He's doing a good job finishing," Sabres coach Lindy Ruff said. "We did a good job of putting it away when we got the key opportunities."
It was a milestone goal for Buffalo: the 10,000th in franchise history.
In the second, Elias and Holik scored for New Jersey before Stafford knotted the score at 2.
After taking a drop pass from Brian Gionta, Elias beat Lalime from the left circle 58 seconds in. Holik put the Devils up 2-1 with his first goal of the season at 5:09. Holik was playing in his fourth game after missing 18 because of a broken pinkie.
"It was a game for us to win and we didn't," Holik said. "I think we played right into their hands. I think we had a great opportunity to continue our streak but we didn't play smart enough."
The Sabres cashed in the first power play of the game as Stafford buried Nathan Gerbe's rebound at 11:05.
In the third, Roy put Buffalo ahead to stay with a 4-on-3 power-play goal from the bottom of the right circle at 8:43.
Vanek connected again at 12:54, and Lalime preserved the win in the closing minutes with saves on Jamie Langenbrunner and Brian Rolston.
It was the second game in two nights for both teams. Buffalo dropped a 2-1 home decision to Toronto on Friday night, and New Jersey beat the New York Rangers 8-5 at home after blowing a four-goal lead.
Notes: The game closed out a three-games-in-four-nights homestand for the Devils. They now hit the road to play Toronto on Tuesday night and Buffalo on Wednesday night. ... Travis Zajac was in the lineup for the Devils. He was considered doubtful after absorbing a big open-ice hit Friday from Rangers defenseman Dan Girardi. ... Weekes last played Nov. 21, a 5-2 win over the New York Islanders. Scott Clemmensen had gone 6-1 since then.
