Under-fire manager Gian Piero Gasperini started Miccoli ahead of Abel Hernandez, and his decision got an instant reward on 13 minutes when the skipper curled in a delightful free kick to open the scoring.
Cheivo pegged their hosts back before half time thanks to a goal from Marco Rigoni.
Miccoli was the star again as he restored Palermo's lead on the hour with a superb solo run and finish.
Palermo's joy quickly turned to nerves minutes later when Franco Brienza was dismissed for a second bookable offence.
Anxiety was lifted for Gasperini ten minutes from time when Miccoli found Luigi Giorgi in space, and he calmly slotted home to make the game safe.
A marvellous Miccoli performance was completed before the final whistle as he completed his hat-trick with an incredible volley from range.
An important result for Palermo lifts the pressure on Gasperini and cements Miccoli's place as a hero on the terraces in Sicily.