The incident, which was missed by match referee Davide Massa and his fellow officials, occurred in the 33rd minute of the Serie A contest at the Stadio Sant'Elia on Sunday.
Destro scored a hat-trick in Roma's 3-1 triumph and has now netted seven goals in his last five matches, but the Giallorossi will be without him for four games after a video review of the clash between the 23-year-old and defender Astori revealed wrongdoing on his part.
Serie A's governing body have handed him a three-game ban and an additional one-game sanction for the number of yellow cards he has accumulated following a caution three minutes before half-time in Sardinia.
Destro has scored 13 goals for Roma this season and is considered to have an outside hope of a place in Italy's squad for the World Cup in Brazil.
However, Destro's transgression may see him fall foul of Italy's code of ethics set by coach Cesare Prandelli, who reserves the right to refuse to call up any player who performs an action he does not approve of.
Rudi Garcia's side will have to rely on the likes of Gervinho, Adem Ljajic and club legend Francesco Totti to provide the goals in Destro's absence as they aim to reduce the eight-point gap to Serie A leaders Juventus.
The capital club have just six games to overhaul that deficit and face the defending champions in the penultimate round of the campaign, which will mark Destro's return from suspension.