Palermo continued to look the stronger of the two sides in the opening 20 minutes, and took the lead when Franco Vazquez fired in from close range for his first Serie A goal.
Verona started to create more chances as the first half progressed, with Luca Toni firing just wide before Vangelis Moras had a header saved.
The hosts were able to get back on level terms just before the break though, as Sinisa Andjelkovic brought Toni down in the penalty area, allowing the striker to convert from the spot.
Palermo started to threaten again as the hour-mark approached, but shots from Dybala, Vazquez and Edgar Barreto all failed to find the target.
Despite looking the more likely of the two sides for much of the second half, Palermo gifted Verona the lead with just over 10 minutes to go, failing to clear a long ball to allow Juanito Gomez to take a shot, but it was Eros Pisano who got the last touch, turning the ball into his own net.
Vazquez desperately tried to snatch a late draw for Palermo, but he sliced one shot high, before seeing his header blocked on the line in stoppage time.
The result moves Verona up to seventh in the table, while Palermo sit 14th.