Stephan El Shaaraway put Milan ahead in the sixth minute of the quarter-final tie before Sebastian Giovinco levelled for the current Serie A leaders and defending champions who will face Lazio in the semi-finals.
El Shaaraway put Juventus ahead when he found the bottom corner from the edge of the penalty area after Kevin-Prince Boateng pulled the ball back. Their lead lasted for only six minutes when Giovinco curled a free kick into the top corner from 20 metres.
Montenegro striker Vucinic struck the decisive blow five minutes into extra time when he exchanged passes with Paolo De Ceglie and poked the ball past Marco Amelia as Juventus avenged their Serie A defeat to Milan in November.
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