The Flyers wrapped up the No. 5 seed in the playoffs and will meet Philadelphia in the first round starting next week.
Buffalo needed a win to stay alive in the post-season race and had a 1-0 lead heading into the final period. Bourdon tied it 6:05 into the final stanza and Read scored the backbreaking goal with 4:21 to play.
The Sabres can still finish in an eight-place tie with Washington, but the Capitals hold a tiebreak edge over Buffalo by virtue of their 37 non-shootout wins this season. Buffalo has only 32.
Ilya Bryzgalov made 25 saves for Philadelphia, which will close out the regular season Saturday at Pittsburgh in a playoff preview.
Buffalo has lost three out of four to fall out of the playoff race for the first time in three years. Ryan Miller gave up two goals on 35 shots.