The Predators had won five games in a row, a streak that has pushed them into contention for the best record in the Western Conference.
Philadelphia had a 2-0 lead heading into the third period thanks to goals by Simmonds in the first and Matt Read in the second. Ryan Suter broke through for Nashville 8:37 into the final period, but Simmonds restored the two-goal lead with 4:41 to play and Claude Giroux scored into an empty net with 1:05 remaining.
The goal for Giroux was his 19th of the season, but his first since Jan. 2.
Bryzgalov finished with 26 saves. Anders Lindback allowed three goals on 27 shots.