It was a tough game for Rudi Garcia's men but they were able to take the three points and boost their chances of a top-two finish.
The striker's first-half goal was followed up by a late effort from Florenzi and it was enough to secure a second consecutive win for the capital club to boost its chances of finishing second in Serie A.
Juventus wrapped up the league title on Saturday but there is still plenty to play for with Roma, Lazio and Napoli all in contention for a top-three finish.
However, it was anything but a simple win for the Giallorossi, as Genoa put them under a lot of pressure during the closing stages in what was a competitive encounter.
M'Baye Niang had the first opportunity of the game when he was found in a great position inside the box but a heavy touch saw the chance pass him by. Moments later, a good through ball presented him with another opportunity but he lost out to Morgan de Sanctis.
Next it was Roma's chance, with Daniele De Rossi unleashing a ferocious shot from distance which flew narrowly wide.
Genoa kept up the pressure, though, with Vasilis Torosidis being forced into a good headed clearance from the back post after a dangerous cross had the side under a lot of pressure.
A great run forward and a cut back from the flank from Gervinho carved out a great opportunity for Roma, with the ball ending up at Radja Nainggolan, whose deflected shot was well saved.
The deadlock was broken shortly afterwards, though, when Doumbia latched onto a defensive error and cut past two defenders before firing home under Mattia Perin.
A good passing move from Genoa 10 minutes into the second half saw Iago Falque through on goal but the winger's effort was blocked by De Sanctis as the visiting side piled on the pressure.
Roma felt it should have had a penalty 65 minutes in when Torosidis was tripped in the box, but the referee refused to entertain the home side's claims and booked the Greece fullback for simulation.
Genoa came close to equalizing with five minutes left when Juraj Kucka beat De Sanctis to a high ball and diverted it towards goal but it was cleared before it could cross the line.
However, the home side was able to put the game to bed in injury time when Florenzi took possession in his own half and burst forward before smashing high past Perin to secure the points.