With its most recent victory over Cagliari on December 22, Venezia is still in 19th place, two points ahead of last-place Monza and four behind Parma.
In the first half, Genoa came close thanks to goals from Nicolussi Caviglia and Kike Perez, but in the second half, Venezia keeper Ionut Radu made fantastic saves to keep out Aaron Martin and Andrea Pinamonti.
However, with eight minutes remaining, Pinamonti broke the tie with a spectacular curling shot, and four minutes later, Genoa's on-loan striker from West Ham, Maxwell Cornet, sealed the game with a cut.