After Florida goalie Sergei Bobrovsky stopped Evan Rodrigues on a breakaway at 1:29, Verhaeghe scored from the left face-off circle.
"It wasn't the way we kind of drew it up, but we'll take the win," Verhaeghe said. "We've got a lot of things to work on still. It's the first game. So we'll take the two points and move on."
Aaron Ekblad scored twice in the third period to tie the game, Aleksander Barkov had two assists, and Bobrovsky made 43 saves for the Panthers, who overcame nine penalties.
Jeff Carter scored his 400th NHL goal, Jake Guentzel scored in his 300th NHL game, and Casey DeSmith made 34 saves for the Penguins, who won one of two their first two games without centers Sidney Crosby (wrist) and Evgeni Malkin (knee). They opened the season with a 6-2 win at the Tampa Bay Lightning on Tuesday.
"To get three out of four points against two really good hockey teams, for me, we should take that as a positive," Pittsburgh coach Mike Sullivan said. "We know we can be better. There's areas, without a doubt, where we can improve. … I just thought our guys competed hard. That's a positive sign."
Ekblad scored goals 1:38 apart after Rodrigues gave the Penguins a 4-2 lead at 8:31 of the third. Ekblad shot into an open net with DeSmith impeded by a scrum in the crease, cutting it to 4-3 at 13:29, before sending a backhand off the crossbar and in to tie it 4-4 at 15:07.
"Crazy game, pretty wild," Ekblad said. "Resilient team. I only say 'Comeback Cats' because we've done it quite a few times in the last few years. We live on that mantra."