The Italy international hit Astori in the face during Roma's 3-1 triumph at Cagliari on Saturday and, although the referee failed to see the action, TV evidence was used to determine the punishment.
Destro, who scored a hat-trick against Cagliari, was booked during the encounter, which has resulted in a further one-match ban.
The 23-year-old will miss the league games against Atalanta, Fiorentina, AC Milan and Catania but will be available for selection for Roma's home game against Juventus on May 9.
A Lega Serie A statement read: "The sporting judge punishes Mattia Destro with a three-match ban for violent conduct in relation to an action that saw the Giallorosso player lash out an arm at Cagliari's Davide Astori.
"Destro is also handed a one-match ban for yellow-card accumulation."
Destro is the Giallorossi's top scorer with 13 goals this season
Roma, coached by Rudi Garcia, are also set to do without injured centre-back Mehdi Benatia for a month, with the Morocco international sidelined with a groin injury sustained at the weekend.
Roma are second in the Serie A standings, eight points behind league leaders Juventus with six games remaining.