Di Carlo had received a vote of confidence from the club's sporting director Giovanni Sartori during the week but his players did nothing to further enhance his job security with a limp display.
In contrast, it was a first win in charge for new Palermo boss Gian Piero Gasperini following two defeats - and he will be toasting Miccoli tonight.
The veteran striker opened the scoring after 13 minutes and restored the hosts' advantage after Marco Rigoni equalised.
Luigi Giorgi made it 3-1 with 10 minutes to go before Miccoli completed his treble shortly afterwards.
Chievo had the better of the opening exchanges before Miccoli struck the opener with a wonderful free-kick which swerved away from Chievo goalkeeper Stefano Sorrentino.
Chievo, who had lost four on the spin coming into the clash, responded well with Boukary Drame seeing a 20-yard drive well saved by Samir Ujkani.
And they grabbed a deserved leveller when Rigoni found space in the six-yard box to stab home from a tight angle.
Rigoni was involved again as Chievo searched for a half-time lead, his corner nodded narrowly over by Paul Papp before David Di Michele also came close.
They may have looked dangerous on the break but Chievo's defence was to prove their undoing.
After Edgar Barreto had seen a shot saved by Sorrentino, Miccoli struck again.
The stocky 33-year-old shrugged off the attention of three Chievo defenders before placing a shot past Sorrentino. His wild celebrations, which included the removal of his shirt, earned him a yellow card.
Palermo's Franco Brienza was also shown a yellow a minute later for deliberate handball but unfortunately for him it was his second of the afternoon and he was dismissed with 28 minutes to go.
Palermo would have been forgiven for trying to preserve what they had but, instead, they poured forward in search of more goals.
And a third arrived after 80 minutes when Giorgi powered a shot into the roof of the net.
And Chievo's miserable afternoon was completed when Jasmin Kurtic's knockdown was superbly converted by Miccoli for his treble.