Sidney Crosby had a goal and an assist for the Penguins (34-15-9), who had won three of four. Tristan Jarry made 25 saves.
"A lot of it boils down to attention to detail, knowing what your job is when we don't have the puck," Pittsburgh coach Mike Sullivan said. "What your role is, what your responsibility is. It takes collective effort to win in this league. When we bring a sufficient level of structure to our game without the puck, I think we're hard to play against."
Duclair gave Florida a 3-2 lead on a power play at 7:02 of the second period with a wrist shot from the slot. His 23rd goal extended his goal streak to three games.
Verhaeghe made it 4-2 at 11:11 of the third with a snap shot on a 2-on-1.
"Everyone played well. Everyone was playing simple hockey," Verhaeghe said. "It was almost like a playoff style. I think these games on the road are really going to be beneficial in the [Stanley Cup] Playoffs."
Crosby got the Penguins within 4-3 with a shot off the stick of Panthers defenseman Aaron Ekblad with 2:09 remaining.
"We're putting ourselves in a tough spot, going down against a really good hockey team like that," Crosby said. "There's no great explanation. We've just got to be better, especially from the start. Give ourselves a better chance. But we've battled back a lot of times in the second period, have good second periods.