Davide Ballardini's men came to life in the second half with Marco Borriello scoring the only goal of the game in the 74th minute to take the Rossoblu out of the bottom three thanks to Roma's win over Siena.
With four games of the season to go, Genoa are level on points with Palermo just above the bottom three, while Chievo are not mathematically safe yet with just a seven-point advantage over the relegation zone.
The fear of losing seemed to stifle Genoa in the first half and even the easiest of passes appeared too much for them.
Chievo did not appear to want to attack much more either, but they had the first effort on goal with Stoian's placed shot from 25 yards pushed away by Sebastien Frey.
Heading into stoppage time, Genoa then almost opened the scoring when a corner found Emiliano Moretti inside the penalty area and his shot on the turn was saved well by the legs of the Chievo goalkeeper.
There was a spring in the step of Genoa when they emerged for the second half and Antonio Floro Flores went close with a direct free-kick.
At the other end, Frey did well to touch Luciano's cross away from the incoming Alberto Paloschi in the 70th minute as the game opened up.
Four minutes later, Genoa's increased pressure was rewarded with Borriello somehow heading a corner into the right-hand corner from close to the penalty spot, although the Genoa striker did not seem to know too much about it.
Marco Andreolli's flick-header from a corner at the other end five minutes later went just wide of the far post with Luca Rigoni unable to get a touch.
Borriello then wasted a perfect chance to double Genoa's lead from seven yards as he dragged his shot wide, but his earlier goal was enough to earn Genoa three crucial points.