The Italian forward is enjoying a successful season and did his chances of a late run into Italy's FIFA World Cup squad no harm after a performance that puts Rudi Garcia's men within five points of Juventus.
Destro's first came just after half an hour thanks to good work from Gervinho and he struck again 10 minutes after the break to finish off a superb counter-attack.
He sprung the offside trap to complete his hat-trick 17 minutes from time and though Mauricio Pinilla scored late on from the spot, it wasn't enough to deny Roma a first win in Cagliari since October 1995 - excluding a 3-0 win they were awarded last year without playing.
Napoli are now 12 points further back in third after Marco Parolo's eighth of the season condemned them to a 1-0 defeat at Parma.
Vincenzo Montella's Fiorentina responded after two winless games by beating Udinese 2-1 and strengthening their hold on the UEFA Europa League places.
They went ahead in the 25th minute through Juan Cuadrado's sixth league goal of the season and Gonzalo Rodriguez doubled the advantage from the spot in the second half.
Meanwhile, Catania's 2-1 defeat to Torino resulted in Rolando Maran's dismissal as coach.
Gonzalo Bergessio gave the home side the perfect start after two minutes but Alexander Farnerud and Ciro Immobile struck late on to condemn the league's bottom side to a fifth straight defeat and it led to Maran's second sacking of the season by Catania.
Lazio moved to within two points of Inter after defeating Sampdoria 2-0 despite playing more than 30 minutes with 10 men.
But it didn't derail Edoardo Reja's side, who clinched their second consecutive win with a Senad Lulic goal.
Lowly Sassuolo sprung a shock by winning 2-0 at Atalanta to end their six-game winning streak.
