Matvei Michkov and Trevor Zegras scored for Philadelphia (42-27-12). Vladar made 24 saves before stopping four more in the shootout.
"'Vladdy' kept us in it, he never quit," Foerster said. "We got the win, and we're all fired up."
Nikolaj Ehlers had a goal and an assist for Carolina (52-22-7), which clinched first place in the Eastern Conference with the point gained by playing past regulation.
"That's a huge accomplishment," Hurricanes coach Rod Brind'Amour said. "It says you had a real good year and you came to play every night. You can't luck into that."
Bradly Nadeau scored, and Brandon Bussi made 21 saves.
"I thought we were good," Carolina forward Jordan Martinook said. "Thought we competed really hard. You know what that game meant to (the Flyers) ... felt like we probably had the better of the chances. Proud of the guys that were playing tonight. We gave it our best and whatever it gets to the shootout, it's flip of a coin. I was happy with it."
Foerster scoring the playoff-clinching goal was a bit of a surprise considering the Flyers didn't expect him to play again after he had surgery on Dec. 15 for an injury to his right arm. He was expected to be out for five months but instead returned on April 2.
"I'll be honest, I didn't think he was going to come back," Flyers coach Rick Tocchet said. "We didn't have a target date, but he was determined. Lot of lonely days for him working out and stuff like that. ... That's a (heck) of a goal. Didn't seem like there was a lot of room there. That goalie looked big, and I'm not sure how he found it. It was a great shot."
Foerster has four points (three goals, one assist) in seven games since returning.
"These guys battled all year for me to come back, so I love them all," he said.
Nadeau gave the Hurricanes a 1-0 lead at 8:41 of the first period when he one-timed Ehlers' pass from behind the net past Vladar's glove.
Ehlers scored on the power play to make it 2-0 at 15:30. Logan Stankoven's pass from along the goal line on the left side of the zone bounced off the skate of Flyers defenseman Travis Sanheim and through the slot to Ehlers, who scored from the right post.
Michkov made it 2-1 at 7:57 of the second period when he scored from the left side after a cross-ice pass from Denver Barkey.