Staal had a hat trick, Malkin scored his first two goals of the series and Crosby added one to help the Penguins cut the series deficit to 3-1. Game 5 is Friday in Pittsburgh.
Marc-Andre Fleury settled down after a shaky first period and had his best game of the series, easily outplaying counterpart Ilya Bryzgalov.
The Flyers led 3-2 when the Penguins reeled off seven straight goals to win the most lopsided game of the series.
The Penguins were the first team to score at least 10 goals in a playoff game since the Los Angeles Kings scored 12 against the Calgary Flames on April 10, 1990, according to STATS LLC.
Claude Giroux, Kimmo Timonen and Jakub Voracek all scored for the Flyers, who blew their chance at a stunning sweep of the 108-point Penguins. Bryzgalov, shaky all series, was at his worst in Game 4. He was yanked for Sergei Bobrovsky after allowing his fifth goal of the game.
There's little chance of a goalie controversy in Philadelphia, though. After earning huge cheers for stopping his first shot, Bobrovsky was worse than Bryzgalov, allowing four goals the rest of the second period.
Fleury made it stand the rest of the way and started to make up for a miserable first three games that saw him allow 17 goals.
Then again, Pittsburgh's offense was so potent, not even Fleury could blow this one.
Long despised in Philadelphia, Crosby scored his second goal of the series on a deflection that tied the game at 3-all in the first. Staal followed with a goal from the slot for a 4-3 lead and the Penguins never looked back.
The Penguins were short-handed in their last-gasp effort at trying to stave off elimination. Forward Arron Asham served the first game of a four-game suspension; and James Neal and Craig Adams served a one-game ban for their actions in Game 3.
Defenseman Paul Martin also sat out with an undisclosed injury.
Trying to stir the pot of an already emotional series, the Flyers showed league disciplinarian Brendan Shanahan's video explanations of the Asham and Neal suspensions.
The crowd was fired up. But so were the Penguins.
Kris Letang shot high over Bryzgalov's right shoulder in the opening minutes of the second for his first goal of the series and a 5-3 lead. Flyers coach Peter Laviolette saw enough and yanked his $51 million goalie.
The goals kept coming at a clip more appropriate for an All-Star game. Staal scored his second, Steve Sullivan and Pascal Dupuis each scored, and Staal slid an easy one through Bobrovsky's legs for the hat trick and a stunning five-goal third.
The Penguins scored four power-play goals in Game 4.
After the two teams combined for a whopping 158 penalty minutes in Game 3, the two teams played under control until the game spiraled out of control late in Game 4.
The nastiness in the series spilled off the ice when Crosby said after Game 3 he didn't like any of the Flyers.
The Flyers whipped up orange T-shirts for the game that read, "Guess What? We Don't Like YOU Either!"
The tabloid Daily News photoshopped Crosby's head on a lion with the headline, "The Cowardly Penguin: Time to Finish Off Sniveling Sidney." One Philadelphia restaurant stuck a sign in its window that read: Crosby: (n). a) a baby penguin. b) slang term for extremely weak.
It was Pittsburgh's captain that got to savor the moment in Philadelphia, though.
For a few minutes in the first, it looked like the Flyers were set to roll toward a four-game sweep. The Flyers scored eight goals in each of the last two games, and scored three on the power play in the first. But the offense dried up and the Penguins took over.
By the time Malkin scored to make it 10-3 in the third, most of the fans had left and the crazed atmosphere from the opening faceoff was a distant memory.
Flyers fans roared when Lauren Hart, the daughter of longtime former Flyers broadcaster Gene Hart, sang "God Bless America," alternating lyrics with Kate Smith, who was on a video image. Smith's rendition of the song has been a rallying anthem for the Flyers since the mid-1970s.
Fans placed roses in the hand of Smith's statue outside the new entertainment complex across from the arena.
They might need a new good-luck charm if the series returns to Philadelphia for Game 6 on Sunday.
Notes: The Flyers are 7-1 in Pittsburgh this season. ... The Flyers have nine power-play goals in the series. ... The Penguins haven't been swept in a series since 1979 vs. Boston.