The Belgium international scored the only goal of the game to help his side to a crucial away victory as they capitalised on Juventus' draw earlier on Sunday
Adem Ljajic missed a spot kick at the half-hour mark, but Radja Nainggolan did open the scoring shortly before the half-time whistle to help Roma to the full three points.
The visitors survived a late scare as the hosts had an injury-time equaliser ruled out for offside, meaning Rudi Garcia's men took full advantage of Juventus' draw against Sampdoria.
The guests started the match the better side of the two and came close to breaking the deadlock after 17 minutes of play when Gervinho was denied by Mattia Perin, before Miralem Pjanic blasted the rebound into a defender.
Genoa were dealt a huge blow at the half-hour mark, however, when the referee awarded Roma a penalty and sent off Perin after the goalkeeper fouled Nainggolan. Ljajic stepped up to take it, but substitute Eugenio Lamanna saved the Serbian attacker's attempt.
Garcia's men did go 1-0 up in the 40th minute. Maicon sent in a dangerous cross from the right and Nainggolan did well to find the net with an acrobatic volley.
Alessandro Matri tested Morgan De Sanctis in the dying seconds of the first half, but he failed to really trouble the Roma goalkeeper.
Pjanic tried his luck early in the second half, yet the midfielder aimed too high and the chance went begging. Gervinho then nearly doubled Roma's lead in the 72nd minute after being set up by Davide Astori, only to see his shot go just wide.
Genoa threatened shortly after via Diego Perotti after the Argentine picked up a loose ball, but the winger's shot went over the crossbar.
The guests kept on pushing for a second goal and Pjanic came close to making it two in the closing stages of the game. The midfielder received a pass from Jose Holebas before unleashing a powerful effort, but his shot went just wide.
The home side thought to have levelled the scoring well into stoppage time, but Tomas Rincon's goal was disallowed for offside.
Roma remain second in Serie A following this weekend's results, but have closed the gap with leaders Juventus to one point. Genoa, meanwhile, sit fifth in the table.