Juraj Kucka and Bosko Jankovic putt Luigi De Canio's side 2-0 up within a quarter of an hour, but Roma were back on terms before half-time thanks to goals from Francesco Totti and Pablo Osvaldo.
Osvaldo's second gave the Giallorossi the lead for the first time the 55th minute and Erik Lamela completed a 4-2 victory late on.
The home side, looking to end a run of three successive draws with a victory, could hardly have imagined a better start to the game.
Kucka put them in front in the seventh minute with former Roma striker Marco Borriello taking some of the credit for a fine backheeled pass which the Genoa midfielder converted with aplomb from the edge of the penalty area.
In the 15th minute, Jankovic made it two at the second attempt.
His first effort, a fine volley from Borriello's cross, came back off the crossbar but, after Kucka returned the ball to him with his head, he made no mistake with another volley beating Maarten Stekelenburg.
Borriello was denied a goal minutes later as Stekelenburg guarded his near post well and Roma finally responded with some attacking of their own.
Totti pulled a goal back in the 28th minute with a low shot just beyond the reach of Sebastien Frey.
Osvaldo, restored to the starting line-up with Daniele De Rossi after both were dropped a fortnight ago, went close to levelling in the 44th minute before making better use of his next chance within seconds.
Ivan Piris sent in the cross for the Italy international to turn and hit an unstoppable volley past Frey.
Osvaldo scored his second and Roma's third in the 55th minute with a header into the top right-hand corner from Alessandro Florenzi's corner.
Behind for the first time in the game, Genoa did not appear to know what to do and Roma gained control of proceedings, adding a fourth in the 83rd minute.
Lamela this time received the ball on the edge of the area and beat Frey with a low shot to effectively finish off Genoa.
Osvaldo almost completed his hat-trick in the 86th minute, but enough damage had already been done as Roma edged up to fifth in the Serie A table, still four points adrift of their city rivals Lazio and eight behind the summit.