The Giallorossi won comfortably Sunday to extend their lead over third-place Napoli to nine points.
The 23-year-old struck early in the first half before Michel Bastos rounded off a comfortable win with a goal deep in injury time.
The first chance of the game fell to Roma and Destro, with the in-form striker narrowly beaten to a ball across the 6-yard box by the outstretched leg of Pedro Mendes.
There was then an opening at the other end as Paolo Cannavaro's strike from a corner kick was deflected behind by Daniele De Rossi in the 12th minute.
And it wasn't long before the deadlock was broken, with the opening goal going the way of the visitors. Radja Nainggolan dispossessed Simone Missiroli in midfield before feeding Destro, who lifted the ball over the onrushing Gianluca Pegolo to make it 1-0.
The Giallorossi remained in control after the break, with Destro going close to doubling his and Roma's lead 53 minutes in when he tested Pegolo with a left-footed drive from the edge of the box.
With 16 minutes remaining the Italian then turned creator as he found Gervinho inside the area, only for the Sassuolo defense to scramble clear before the forward could get his shot away.
Substitute Bastos made it 2-0 with a goal in the sixth minute of injury time to leave Roma in second place and nine points clear of Napoli in third.