Peverley scored his 10th and 11th goals of the season and his second tally, coming with 13:27 left in the second period, drove goaltender Tomas Vokoun from the game.
Ondrej Pavelec made 33 saves for Atlanta, limiting Florida to a single goal over the final two periods. Atlanta won for the third time in four contests.
Vokoun was removed after giving up three goals on 15 shots. Scott Clemmensen stopped all 16 shots he saw the rest of the way.
Niclas Bergfors also scored for Atlanta, producing the game's first goal with 7:08 to go in the first period.
Florida had a two-game winning streak halted.