Paulo Fonseca's side pull one point ahead of Atalanta who drop to fifth before their game Monday at Serie A leaders Inter Milan.
Mancini jumped highest to turn in a Lorenzo Pellegrini corner after 24 minutes in the Stadio Olimpico, beating Genoa second choice 'keeper Federico Marchetti who replaced Mattia Perrin.
Roma extended their unbeaten run against 13th-placed Genoa to 14 games, to sit nine points behind leaders Inter Milan. The three points were a boost for the capital side days before hosting Shakhtar Donetsk in their Europa League last 16, first leg tie.