Genovese began studying music and accounting at the University of Rosario, but soon abandoned accountancy, and in 2001 he began studying at the Berklee College of Music.[1] He graduated in 2003.[1] His first album, Haiku II, was released the following year and was followed by Unlocked in 2008, but Genovese later talked them down, stating that they were "just a way to document where I was at the time".[1] From 2005 he recorded and toured internationally with bassist and vocalist Esperanza Spalding.[1]

A reviewer for The New York Times commented on Genovese's 2013 album, Seeds, that, "by refusing to privilege one historical style over another, he strengthens his claim as a polyglot".[2] Down Beat observed that Genovese's compositions for the album "share an exploratory nature, whether the new terrain in question is a marriage of electronic and acoustic sounds, an unlikely use of chromatic scaling or the successful juxtaposition of otherwise disparate ideas."[3]

Wayne Shorter & Leo Genovese, soloist won a Grammy Award for Best Improvised Jazz Solo for the Wayne Shorter composition "Endangered Species" for which Leo accompanied him on the album Live at the Detroit Jazz Festival.[4] Genovese won the DownBeat Critics Award for Rising Star – Piano in 2023