Jordan Staal had a short-handed goal and an assist for Pittsburgh, while Marc- Andre Fleury recorded his 30th win of the season with a 27-save effort.
Jaromir Jagr scored two goals in an 18-second span during the first period for Philadelphia, which has dropped six of eight.
"We're behind our two goalies and we have confidence in them," said Philadelphia forward Claude Giroux. "Bad goals are going to happen, but they're two good goalies and we know we have to be better in front of them."
Help may have come in the form of another trade.
The Flyers acquired defenseman Pavel Kubina from the Tampa Bay Lightning on Saturday in exchange for forward Jon Kalinski, a conditional second-round draft pick previously acquired from the Florida Panthers and a fourth-round pick in 2013.
Meanwhile, defenseman Nicklas Grossman made his Flyers debut after he was acquired from Dallas on Thursday for two draft picks.
Jeffrey broke a 3-3 tie when he pounced on the rebound of a Cooke shot 37 seconds into the third. The Penguins grabbed a two-goal lead when Dupuis' shot from the left side of the net caromed in off Bobrovsky's left skate at 2:09.
Neal backhanded his own rebound at 9:02 for his 30th goal of the season and a 6-3 Pittsburgh lead.
Wayne Simmonds capped the scoring with 19 seconds to play.
Cooke beat Bryzgalov on the short side with a wrister from the left circle 3:17 into the game.
The Flyers tied the game while the teams skated 4-on-4. Jagr carried the puck down the right wing on a 3-on-2, looked off Philadelphia defenseman Andrej Meszaros as Danny Briere drove the net and fired a shot past Fleury.
Staal was called for holding on the ensuing faceoff, and Jagr, who missed the previous game with the flu, buried another shot top shelf at 15:34.
Pittsburgh, though, regained the lead in the second period with consecutive short-handed goals, including one while down two skaters.
Bryzgalov fanned on Staal's shot with his glove at 15:14, then Cooke won a race to a loose puck during a 5-on-3 power play that bounced over Kimmo Timonen's stick and flicked it past the Philly netminder at 16:57.
"Our focus is to kill [the penalties] first," Cooke said. "Obviously when given the opportunity or given the chance to go on the offense, we want to do that."
The Flyers caught a break and tied the game with 1:19 remaining in the middle stanza. Pittsburgh defenseman Deryk Engelland wiped out in front of Fleury and Eric Wellwood potted his first career NHL goal.
The last 5-on-3 goal against the Flyers was on March 9, 1993 at the New York Islanders when Benoit Hogue scored into an empty net at 19:59 of the third period. The last 5-on-3 goal allowed by the Flyers that was not an empty-net goal was December 28, 1979 at Winnipeg, scored by Jude Drouin at 4:52 of the first period...Pittsburgh's Evgeni Malkin picked up an assist to push his point streak in Philly to 10 games (six goals, eight assists)...Flyers forward James van Riemsdyk returned to the lineup after missing a month with a concussion...The Penguins improved to 6-2-1 in their last nine road games...The Flyers fell to 4-10-1 this season in afternoon games...Wellwood became the eighth Flyers rookie to score a goal this year.