Juve defeated Atalanta 1-0 on Wednesday night.
Elkann told Mediaset: "It's great to be here and win the Italian Cup. Challenging match, I'm happy for the young guys who are at Juve and who have never won yet.
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"Winning helps to win, we take this evening as a good moment for Juventus and we look to the future with optimism and hope.
"Juve has excellent young players, as they demonstrated this evening."