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 Argentina-born Italy striker, who has had a fractious relationship with a section of the Roma faithful despite an impressive scoring record since arriving from Espanyol in summer 2011, celebrated his first and third goals by putting his hand to his ear and then pointing to his name on the back of his shirt.
The 27-year-old told the Corriere Dello Sport: "I want to thank the fans who have supported me.
"There's always a section that unfortunately still goes against me, but I really want to wear this shirt and try to prove myself. Sometimes I can, sometimes not.
"But the desire is always there and now, fortunately and thanks to my team-mates, I scored three goals. I hope that from now on everything is a little more quiet."
The result lifted Aurelio Andreazzoli's side to fifth in the standings, while Giuseppe Iachini's Siena slumped to 19th following wins for fellow strugglers Palermo and Genoa.