Vanek's tally late in the second period forged a 2-2 tie and his assist on Tyler Myers' score with 4 minutes left in the game put the Sabres ahead to stay.
Steve Ott and Cody Hodgson also found the net while Jason Pominville distributed three assists as the Sabres began their quest to improve on last season's performance, in which they missed the playoffs for third time in five years.
Ryan Miller turned aside 27-of-29 shots for Buffalo.
The Flyers, meanwhile, fell for the second time in two days to start the season despite goals from Sean Couturier and Claude Giroux.
Ilya Bryzgalov was touched for four goals on 40 shots in the loss.
Celebrity Families of 2014 [PHOTOS]