Juve, who have seven games remaining, can reopen their eight-point gap with victory over Livorno on Monday night, but nerves will be frayed in Turin after Roma demonstrated their determination to snatch the Scudetto.
Destro scored once in the first half and twice after the break to reach 13 goals for the season before Mauricio Pinilla gave 15th-placed Cagliari some reason to be cheerful by blotting Roma's clean sheet with a penalty at the death.
Roma were awarded a 3-0 walkover victory instead of fulfilling this fixture last season after Cagliari president Massimo Cellino decided to flout a behind-closed-doors mandate.
The capital club were able to play in southern Sardinia this time, though, and needed to get the goals and three points under their own steam if they were to keep chasing the title.
Cagliari threatened first, Daniele Dessena feeding Pinilla to head just over the crossbar in the sixth minute.
The Chilean striker's next effort ended up in Morgan de Sanctis' gloves before Roma hit back through Maicon, whose long-range effort sailed too high.
Nene went dangerously close to opening the scoring in the 23rd minute, but his free-kick shaved the bar.
Eight minutes later, Destro finished off a swift counter-attack started by Gervinho by clipping home with no fuss.
Home captain Daniele Conti had a late opportunity to level shortly before the break, but his free-kick spun off the wall for a corner and Roma retained their lead.
The Giallorossi went all out for the second goal when play resumed and, after former Cagliari midfielder Radja Nainggolan rattled a shot into Vlada Avramov's gloves, he tried to beat the Serbian from distance.
Roma doubled their lead in the 56th minute and Nainggolan was heavily involved as he served Destro with a cross from deep that the Italy striker slotted past Avramov.
Cagliari went close to cutting the deficit before the hour mark when Pinilla tried to beat De Sanctis with his right boot and he lived to regret his miss as Roma soon pulled further ahead.
Destro's hat-trick goal arrived in the 73rd minute. This time the one-time Siena forward swept up Alessandro Florenzi's assist from the left to beat the beleaguered Avramov again.
The chase might have been beyond Cagliari, but they battled bravely to make the scoreline respectable and Victor Ibarbo stung De Sanctis' palms before Danilo Avelar worried the keeper from distance.
In the 89th minute, Cagliari got their consolation goal.
After being fouled by Mehdi Benatia, former Hearts striker Pinilla happily rifled the ball past De Sanctis from the spot.