Conte delighted as Juventus beat AC Milan in Coppa thriller
Stephan El Shaarawy put Milan ahead on six minutes before Sebastian Giovinco struck an equaliser just six minutes later.
The match went into extra-time before Vucinic's winner on 95 minutes.
Juve coach Antonio Conte said afterwards: "Our pride is strong, there was desire on our part and we used a lot of players who had played less. It was a great response by all and we just wanted to go through."