An early Dusan Vlahovic strike at Stadio Olimpico was enough for I Bianconeri to end their three-year trophy drought and win a record-extending 15th Italian Cup in club history.
Atalanta dominated possession but failed to manage a single shot on target.
Allegri became the first coach to win the Italian Cup five times, breaking a tie with Sven-Goran Eriksson and Roberto Mancini.
“We qualified for the Champions League next season and we’ve won the Italian Cup, so we’ve achieved our goals for this season,” – Dusan Vlahovic