Girgensons' overtime winner was his first goal since Feb. 24.
Johnson made 30 saves in the victory as the Sabres improved to 3-6-0 on the season. They'll play the Pittsburgh Penguins on Thursday.
The Flyers, who fall to 4-2-2, salvaged a point when Brayden Schenn tipped a Claude Giroux shot past Johnson to tie the game with 53.2 seconds remaining. They will look to get back on the winning track against the New Jersey Devils.