Mike Hoffman scored his first goal of the season for the Sharks (2-13-1), who have lost three straight. Mackenzie Blackwood made 27 saves.
"There were a lot of good signs overall throughout the game," San Jose coach David Quinn said. "But against a team of that caliber, you've got to make sure that you're thorough in all areas for 60 minutes."
Tomas Hertl tied it 2-2 at 6:11 of the third period when he redirected in a centering pass from Fabian Zetterlund on a power play.
Luke Kunin put San Jose ahead 3-2 during a delayed penalty at 7:21. He scored five-hole with a backhand after Mikael Granlund's shot from the point caromed to him off the end boards.
Verhaeghe tied the game again 3-3 at 10:06 with a one-timer from the right circle on a power play.
"We had the lead in the third period, and then we take a penalty (and) they take advantage of it," Hoffman said. "We kept fighting and we never gave up. We got a couple of chances to tie it, but that wasn't enough."
Kevin Stenlund put Florida back in front 4-3 at 11:09 when he redirected a point shot from Uvis Balinskis five-hole on Blackwood.
"We know we can score at any time," Stenlund said. "We're confident. We've just got to keep working. We did that and we won."