Daniel Osvaldo scored the only goal of the game in the third minute to end his three-month goalscoring drought for the club.
Rodrigo Palacio struck the crossbar for Genoa in the 71st minute.
The win followed Roma's 1-0 win at Palermo and saw the Giallorossi close the gap on third-placed Lazio to four points.
Lazio hold the last Champions League qualifying spot with 10 games remaining.
Genoa, meanwhile, have not won away since December 10, and have dropped to 14th in the table.
The visitors went into the game at the Stadio Olimpico without six players due to injury, including defenders Cesare Bovo, Andreas Granqvist and Luca Antonelli.
Roma captain Francesco Totti and midfielder Miralem Pjanic were also hurt and did not play.
Osvaldo put the hosts in front almost from the off.
He controlled Leandro Greco's long cross in the area and hit a diagonal shot past Genoa goalkeeper Sebastien Frey.
Roma continued to push forward and almost doubled their lead but Frey kept out Greco's attempt from 30 yards.
The home side wasted a good chance midway through the first half when Osvaldo beat the offside trap and surged inside the area, only to fire wide.
Genoa threatened in the 32nd minute as Fernando Berluschi got a touch on Palacio's cross but his effort was kept out by Roma goalkeeper Maarten Stekelenburg.
On the stroke of half-time, Giandomenico Mesto's header from Palacio's cross was saved by Stekelenburg.
Roma showed more intent after the re-start and before the hour mark, Fabio Borini tested Frey with a hard drive from outside the area which the keeper saved.
Daniele De Rossi then fed substitute Bojan in the area but he hit the ball just wide as another opportunity went begging.
Roma were fortunate not to concede when Palacio's volley from substitute Cristobal Jorquera's dangerous cross hit the woodwork with Stekelenburg already beaten.
Genoa continued to threaten but Roma held on for three valuable points.