Osvaldo netted twice in the first half, with Erik Lamela on target in between, and then completed his treble midway through the second period.
The result lifted Aurelio Andreazzoli's side to fifth in the standings, while Giuseppe Iachini's side slumped to 19th following wins for fellow strugglers Palermo and Genoa.
The writing was on the wall from the early exchanges for the Robur, Osvaldo heading over the bar from Francesco Totti's right-wing cross in the fourth minute.
Osvaldo fired wide six minutes later as the hosts continued to push for the opener, which duly arrived on 14 minutes.
Lamela fed Osvaldo in the six-yard box and the Argentina-born Italy striker made no mistake.
Two minutes later it was 2-0, Lamela lashing a fine first-time shot into the top-right corner following superb work from Totti.
Siena mustered their first chance after 24 minutes but Roma goalkeeper Bogdan Lobont was equal to Innocent Emeghara's effort.
Osvaldo almost extended the Eternal City club's advantage two minutes later but, after shaking off the attentions of defender Massimo Paci, saw his right-footed shot hit the post.
He was not to be denied four minutes before half-time, though, beating Gianluca Pegolo at his near post from Alessandro Florenzi's assist.
Lamela should have grabbed his second of the game five minutes after the restart but fluffed his lines when teed up by Osvaldo.
Lobont was called into action to preserve his clean sheet in the 62nd minute, keeping out a decent stab from Emeghara which had looked destined for the bottom-left corner.
And, five minutes later, the rout was complete. Pegolo saved Totti's 20-yard attempt but Osvaldo seized on the rebound to seal his hat-trick.
It could have got even better for the hosts in the closing stages but Osvaldo was denied a fourth by Pegolo and substitute Simone Perotta's fierce 30-yard drive flew narrowly wide.