Despite their early setback, Genoa took the lead just two minutes later as the ball broke kindly inside the box to Diogo Figueiras, who fired a low drive past Vid Belec and into the far corner.
The Rossoblu continued to pose a threat and Armando Izzo headed narrowly wide, before Tino Costa volleyed off target from a corner.
Darko Lazovic then almost grabbed a second on the stroke of half time with a powerful drive from a tight angle that was pushed behind at the near post.
Genoa were made to pay for not taking their chances as in the 57th minute Marco Boriello was found inside the box by Antonio Di Gaudio, before drilling a low strike into the far corner.
The goal provided a major boost in confidence for the visitors and with 10 minutes of the match left they scored a second when Boriello played in Cristian Zaccardo to slot in at the near post.
Tomas Rincon almost grabbed an equaliser for Genoa moments later, but his effort towards the top corner was tipped over by Belec as the visitors held on to record a precious win in their battle against relegation.