Pablo Osvaldo fired home the game's only goal after less than three minutes but despite always looking the better side, Roma could not find the second goal to kill off the game.
As the game progressed, the jitters increased and Genoa piled on the pressure in the closing stages in search of an equaliser.
However, Luis Enrique's young side, without injured talisman Francesco Totti, held on for the win and while they remain in sixth position, they are now just four points off third placed Lazio and three behind Napoli and Udinese who are fourth and fifth respectively.
Osvaldo, back in the team after suspension, put the hosts into the lead right out of the blocks with a fine opener.
The former Espanyol man was found beyond the back post by a lovely Leandro Greco chipped ball, he then cut inside onto his right foot and drilled a low shot into the corner of the net for his eight goal of the season.
Roma's crisp passing helped them dominate for the majority of the first half and Sebastien Frey had to be alert to push away decent drives from Greco and Daniele De Rossi.
Osvaldo also had a couple of chances to add to his goal, firing wide in the 22th minute after springing the offside trap, and then fizzing an effort just over seven minutes later when he again had cut inside onto his right foot.
Genoa were limited in the first half but Giandomenico Mesto had a half-chance in the 33rd minute when Roma failed to properly deal with a Rodrigo Palacio cross.
The visitors were better in the second half but that was as much because Roma started to give the ball away cheaply as anything they were doing themselves.
They did have one superb chance in the 70th minute when a ball was played over the defence for Palacio and after controlling the ball, the in-form Argentine striker looked all set to score but instead managed to hit the crossbar from just six yards out.
Possession was much more even in the second period and Roma started defending deeper and deeper.
To be fair to the
It was far from vintage stuff from Roma, but a season that started so poorly is finally starting to gather some momentum and they are right in the hunt for European places now with a big match away at AC Milan coming up next week.
Genoa on the other hand remain in 14th position – with seven points between them and 18th place Lecce there should be no late season drama for Pasquale Marino's men although they will still need a few more points to secure their Serie A status for next season.