The Flyers scored three shorthanded goals in the game to force overtime, but the Stars got the extra point with Joe Pavelski's game-winning goal in overtime in a 5-4 loss on Saturday night.
Jason Robertson's first of two goals came with 1.3 seconds remaining in the first period on the power play, setting up a 4-1 Flyers loss to the Dallas Stars on Thursday night.
The Flyers begin the final five games of the season on Thursday night against the Dallas Stars, who currently sit in a three-way tie atop the Central Division.
Jacob Peterson scored the eventual game-winning goal at 16:35 of the third to put the Dallas Stars ahead in a 3-1 loss that was the Flyers 12th straight on Monday night at Wells Fargo Center.
The Flyers will face the Dallas Stars on Monday night while on the cusp of two historical moments. Keith Yandle with tie Doug Jarvis' consecutive games record at 964, while the Flyers would match a franchise-record winless streak of 12 with another defeat.
The result was a 5-2 loss for the Flyers to the Stars, who got the only goal of a first period while getting heavily out-shot, then took control for the remainder.
Corey Perry scored his second goal of the game on a rebound at 9:23 of the second overtime to give the Stars a 3-2 win in Game 5 and force a Game 6 on Monday night.