Verona started to look the more likely as the first half progressed, but Luca Toni's header from a promising Leandro Greco cross flew just high of the crossbar, before a solo effort by the striker resulted in a wide finish.
The home side came close again just before the interval as Tino Costa attempted to catch the Verona keeper out with a long-range strike, but it flew just wide of the post, while a drive from just outside the box from Serge Gakpe moments later went the same way.
Genoa continued to enjoy the majority of the possession early in the second half, and opened the scoring just before the hour mark as Leonardo Pavoletti pounced on a loose ball in the box, jabbing it past Rafael just a minute after coming off the bench.
Rincon had a chance to quickly double the lead for the home side when Issa Cissokho delivered a cross, which Rafael initially punched away, leaving the Genoa man to drive the ball at the goal from the edge of the box, watching it go just wide.
Rincon did help Genoa find a second inside the final 15 minutes though, delivering a cross for Gakpe to clinically nod into the back of an empty net.
The visitors tried to make it close late on, with Toni curling the ball in from out wide, narrowly missing the target, before Giampaolo Pazzini's header from a Jacopo Sala cross late on went over the crossbar.
The result gives Genoa their first points of the season, moving them up to 10th in the table, while Verona sit 15th.