After disappointing defeats against Atalanta and city rivals Lazio, Luis Enrique's men have now won two on the bounce to close to within four points of third place
The Argentine-born Italy international's strike arrived just two minutes in, which deceived those in attendance into thinking that they were in for an entertaining evening.
However, Roma failed dismally to build on their dream start and were ultimately somewhat fortunate to escape with all three points, with Genoa forward Rodrigo Palacio having struck the crossbar midway through the second half.
Shorn of the services of captain Francesco Totti and midfielder Miralem Pjanic through injury, Roma lacked creativity and dynamism throughout and it was Genoa who actually started with greater purpose.
However, it was Roma who took the lead, with Osvaldo controlling a long ball forward from Leandro Greco and then cutting inside onto his right foot before finishing smartly into the bottom right corner of Sebastien Frey's goal.
The hosts only threatened sporadically thereafter, though Osvaldo did squander a great chance to double his side's advantage after being put through by Rodrigo Taddei.
Consequently, Genoa grew in confidence as the game wore on and just moments after Miguel Veloso had drawn a decent save out of Maarten Stekelenburg with a long-range effort, Palacio chested down a cross into the area from Cristobal Jorquera only to then flick the ball onto the crossbar and away to safety.
The Rossoblu pressed forward in search of an equaliser in the closing stages, leaving themselves wide open to the counterattack, and Bojan Krkic went close on a couple of occasions for the hosts.
As it was, though, Osvaldo's goal proved sufficient to propel sixth-placed Roma to within four points of third-placed Lazio, meaning their hopes of securing a Champions League place remain very much alive.