Paloschi put the hosts ahead in the fifth minute and that goal looked as if it would be the winner for the much the match.
However, in a dramatic finale, Alberto Gilardino equalised two minutes from time for Genoa, only for Paloschi to convert his second spot-kick on the final play of the game after substitute Dejan Lazarevic was fouled by Marco Motta in the area.
Chievo's victory, which brought an end to Genoa's four-match unbeaten run, lifted the Verona outfit to 16th place and three points clear of the bottom three.
Genoa dropped to 11th with the defeat.
Chievo had mustered just one win in their past 11 fixtures before Sunday's game but they took an early lead after Moussa Konate pulled down Adrian Stoian in the box.
Paloschi struck his penalty towards the bottom right corner of goal and out of Genoa goalkeeper Mattia Perin's reach.
Genoa almost equalised a minute later but Motta's close-range header from Stefano Sturaro's cross went wide, and then shortly after, Sturaro's powerful free-kick flew over the bar as the visitors got closer.
In the 20th minute, Sebastian De Maio met a corner kick with a powerful shot from 10 yards that tested Chievo goalkeeper Michael Agazzi's reflexes before Gilardino had a goal correctly ruled out for offside.
Chievo weathered that storm though and almost doubled their lead before half-time when Perparim Hetemaj's goal-bound attempt from close range was saved by Perin.
Genoa came back at Chievo after the break and almost pulled level with the contest nearly an hour old.
Agazzi could not hold onto substitute Ricardo Centurion's header and Ioannis Fetfatzisdis pounced, only to strike the rebound wide.
A minute later, Gilardino failed to find the target with a header from Paolo De Ceglie's cross as another chance went begging.
Genoa finally managed to get on level terms just before the end, though, when Gilardino's close-range header from Sturaro's cross beat Agazzi.
However, Paloschi had the last say when he scored his second from the spot.
The celebrations started for Chievo after the ball went past Perin and into the bottom right corner of goal.