Catania were the dominant side for most of the first half, with Pablo Barrientos, Alejandro Gomez, Francesco Lodi, Gonzalo Bergessio and Sergio Almiron all testing the Genoa backline inside the first 25 minutes.
However, much against the run of play the visitors took the lead on 26 minutes when Ciro Immobile set up Juraj Kucka to slot home from close range with his left foot.
That spurred on Catania even more, as Almiron and Bergessio tested Genoa goalkeeper Sebastien Frey near the half-hour mark.
The home side eventually made it 1-1 when Bergessio scored from close range in the 66th minute.
Two minutes later Bergessio made it 2-1 to Catania when he headed home Alejandro Gomez's corner.
There was yet more drama to come at Stadio Angelo Massimino, as Bosko Jankovic netted from outside the Catania area to make it 2-2.
Then with just two minutes of normal time left on the clock, Lodi scored from a free kick for Catania to wrap up a 3-2 victory.