Defenseman Tyler Myers scored the go-ahead goal with 5:03 left after the Sabres overcame a 2-1 second-period deficit. Cody Hodgson and newly acquired forward Steve Ott also scored for the Sabres. Captain Jason Pominville set up two goals and Ryan Miller made 27 saves in helping Buffalo improve to 6-1-1 in its past eight season openers.
Vanek became the first Sabres player with five points in a game since Drew Stafford had three goals and two assists in January 2008.
Claude Giroux, with his second goal of the season, and Sean Couturier scored for the Flyers, who were coming off a 3-1 season-opening loss to Pittsburgh on Saturday. This marks just the 10th time Philadelphia has opened 0-2.
The Sabres scored the game's final four goals, with Vanek capping the run with an empty-netter in the final minute.
Myers put the Sabres ahead for good with Flyers defenseman Luke Schenn off for tripping. The Sabres worked the puck around the top of the zone to Myers at the left point. Myers got a shot off that sneaked in through the legs of Ilya Bryzgalov, who appeared to be caught leaning left with Vanek parked in front.
Hodgson scored 1:16 later to make it 4-2. Vanek was stopped on a breakaway, and Hodgson followed to stuff in the bouncing rebound in the crease.
The Sabres converted 3 of 6 power-play chances. The Flyers' penalty-killing unit has now surrendered five goals in nine chances after giving up two power-play goals in the 3-1 season-opening loss.
After Ott opened the scoring with a first-period power-play goal, the Flyers responded with two goals in a 2:34 span early in the second period.
Couturier tied it by tipping in Kurtis Foster's shot from the blue line. Giroux then scored by blasting in a shot from the left circle that beat Miller over the left shoulder.
Vanek tied it with 4:06 left in the second period when the Flyers were down two players. Just as Buffalo's two-man advantage was coming to an end, Pominville sent a pass from deep in his own zone to Stafford parked just outside the Flyers' blue line. Stafford slid a pass into the middle, where Vanek got to it in stride and broke in alone to beat Bryzgalov.
The Sabres were without third-line forward Ville Leino, who is listed as day to day with a lower body injury after he was hurt in practice Friday.
Before the game, Sabres players, coaches and team owner Terry Pegula were in the First Niagara Center lobby to greet fans when the gates opened. They handed out programs, signed autographs and posed for pictures.