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Allegri became the first coach to win the Coppa Italia five times after Sven-Goran Eriksson and Roberto Mancini lifted the trophy on four occasions.
The 2-1 victory over Atalanta also handed a first trophy for coach Andrea Pirlo.
Paulo Dybala and Danilo missed Juve's first two efforts in the shootout while Napoli converted all their attempts to win the trophy for the sixth time.
Results 1-3
