Philadelphia's Wayne Simmonds scored with less than a minute left in the third period to tie the score at 3. James van Riemsdyk and Matt Carle the Flyers' other goals, and van Riemsdyk also had two assists.
It was Philadelphia's third game in three nights after splitting a home-and-home with Toronto. The Flyers will make it four in a row Friday night when they play the Red Wings in Detroit.
Helm caught the Flyers flat-footed while killing off a penalty when he stole the puck from Graydon Coburn at center-ice, skated in and sent the puck over Mike Leighton's shoulder midway through the first period.
Filppula made it 2-0 with about 4 1/2 minutes left in the second, and van Riemsdyk snapped a rebound past Red Wings starter Ty Conklin 2 minutes later to pull the Flyers within one.
White regained Detroit's two-goal lead with a power-play score about 6 1/2 minutes into the third period.
Carle pulled Philadelphia to 3-2 with 2:45 left, and Simmonds tied it with 53 seconds to go.