Darren Helm, Valtteri Filppula and White scored in regulation for the Red Wings.
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.
As Simmonds took the first shot in the shootout, a banana came flying into his path as he skated in on Detroit goalie Jordan Pearce.
"I don't know if it had anything to do with the fact I'm black," the Toronto native said. "I certainly hope not. When you're black, you kind of expect (racist) things. You learn to deal with it."
Simmonds acknowledged the banana rattled him, and his only thought was to get off a shot - which went in for the Flyers' only score in the shootout.
"It shocked me and knew I had to keep going and get a shot off," he said. "It was certainly unusual."
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 4½ 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 6½ minutes into the third period.
Carle pulled Philadelphia to 3-2 with 2:45 left, and Simmonds tied it with 53 seconds to go.