"If you know ... they are tired, you've got to keep on going after them, make sure they make a mistake," said Fleischmann, the beneficiary of a late miscue. "They did, and that's why we won the game."
Jacob Markstrom stopped 33 shots while flirting with his first career shutout, and Filip Kuba had two assists for the last-place Panthers, who scored the first three goals and held on to snap their six-game slide.
Markstrom was 2:33 from the shutout when Jordan Staal scored on the power play for Carolina.
Dan Ellis stopped 40 shots for the Hurricanes, who entered atop the Southeast Division but lost their fourth straight and dropped a shootout to the Rangers in New York the night before.
"Our effort and our compete was there. I thought we looked a little worn-down but we were staying with our structure and our system," captain Eric Staal said. "It was a 0-0 game going into the third period, and some mental mistakes and it ends up in the back of your net."
An untimely turnover by Bobby Sanguinetti -- one of three Carolina defensemen who were injured in the first two periods -- wound up turning into the game's first goal.
His misplay near Ellis sent the puck to Tomas Kopecky, who set up Fleischmann's wrister to make it 1-0 at 7:39.
"We could tell they were getting tired, minute by minute, and our forechecking, that's what got us the first goal," Fleischmann said. "And then we just, I would say, played better than them and better physically."
Goc scored on a tip-in on a 5-on-3 power play at 12:47, and Matthias -- who was ill with flu-like symptoms, slept all day and was a candidate to be scratched -- added a goal with the man advantage at 13:51.
"I don't want to miss any games," Matthias said. "I want to play and get us out of this."
That was plenty of offense for the Panthers behind Markstrom, the second-year goalie who entered 1-5-1 this season with a goals-against average of 3.29.
He certainly found a groove against a Carolina offense that, until the final minutes, had sputtered. Since blowing a 2-0 lead in a 3-2 loss to division rival Washington on March 14, the Hurricanes have managed just three goals in their next three games.
Their power-play unit, the second-worst in the NHL, struggled against the Panthers' league-worst penalty-kill, missing their first three chances before Jordan Staal's late goal -- Carolina's first with the man advantage since March 9.
Their frustration boiled over when Alexander Semin skated to the bench after his shift ended and whacked his stick on the boards.
"I thought actually at times tonight, it actually looked better than it has in a month," coach Kirk Muller said. "There's just no results. You've just got to let them sort through it and get one goal, and I think their confidence level will grow on it and then it'll just come natural again."
Semin's frustration came during a critical 4-minute power play after Scottie Upshaw received a double-minor for high-sticking a bloodied Tim Gleason -- one of those three injured Hurricanes defensemen.
Justin Faulk suffered an apparent right leg injury during a collision behind the boards with Jack Skille moments before Gleason was hurt and didn't return, while Sanguinetti appeared to take a skate to his face with 7:03 left in the first but was back for the second. Gleason returned for the final 2 minutes of the period.
Muller said Faulk would be evaluated Wednesday.
"When we lost Faulk, more than anything, going to five (defensemen), back-to-back games, really hurt our D going into that going into the third," Muller said. "It kind of starts to put some holes in the game going into the third period."
Both teams certainly had plenty of chances early: Semin clicked a pass off the shaft of a Florida player's stick on its way across the crease, Kevin Westgarth rang the post, and Drayson Bowman had a breakaway yet couldn't get a shot off on Markstrom after the puck rolled off his stick.
The Panthers -- who won for just the seventh time since beating Carolina in the opener -- couldn't get anything past Ellis for two periods, either, despite peppering him with a handful shots in roughly a minute of the first.
The teams split the season series 2-2. ... Carolina's record against the Panthers at PNC Arena is 28-7-4. ... Westgarth dropped LW Eric Selleck during a brief fight with 2:58 left. Selleck received an instigator misconduct. ... Before the game, the Panthers recalled C Quinton Howden from San Antonio of the AHL, then scratched him.