Riccardo Meggiorini had the game's first chance after some poor defending allowed the hosts to find the forward with a long throw, but his strike flew into the side-netting.
Aside from that chance, it was mostly one-way traffic for the home side, who were able to break the deadlock in the 36th minute when Paloschi beat his markers to slot the ball into the bottom-left corner.
The visitors started the second half with intent, but were unable to stem the Flying Donkeys' forays forward as Bizzarri had to be at his best once again to thwart an Ervin Zukanovic header.
Paloschi then had the chance to double his side's lead from the spot when Sinisa Andelkovic handled in the area, prompting the referee to give the hosts a penalty, but the 25-year-old sliced over the bar.
Palermo were unable to make the most of the reprieve, going down to 10 men when Andelkovic was shown his marching orders for a second yellow card four minutes from time.
The result sees Chievo climb to 14th in the Serie A table, while the Sicilians stay in 11th place.