A scrappy opening to proceedings produced just one clear chance, with Roma's Radja Nainggolan firing wide of the post after nine minutes of football.
Genoa's best chance of the half fell to Emanuele Calaio after 20 minutes, but the 32-year-old forward could only fire wide of the goal from inside the box as both sides continued to exchange periods of possession in a slightly untidy contest.
Roma improved as the half progressed, however, and forced four chances in quick succession before the 30-minue mark. First, Genoa goalkeeper Mattia Perin twice denied Adem Ljajic, before Nainggolan and Mattia Destro saw efforts move high of the crossbar.
Roma continued to dominate in the early stages of the second period, with Nainggolan having another long-range effort blocked following smart work from Ljajic.
Genoa secured all the points late on, however, when Ioannis Fetfatzidis found the bottom corner from just inside the box.
The defeat was Roma's fifth of the season, with the runners-up finishing 17 points behind champions Juventus.
Genoa, meanwhile, ended the campaign in 13th position.