The Milan side went ahead in the sixth minute through Stephan El Shaarwy, and six minutes later, Sebastian Giovinco equalised for the Old Lady with a free kick Wednesday night, reports Xinhua. Then they had to wait until five minutes into the extra time when Vucinic scored on a fast counter-attack.
With the win, Juventus head coach Antonio Conte insisted he received "a great response from everyone".
"It is always a tough game between these two great teams and clubs who have won so much," Conte said.
"Our hunger was huge and I received a great response from everyone, especially as I included many players who hadn't featured in a while.
"I told them that what happened against Sampdoria must never happen again," he added, referring to Sunday's 2-1 defeat to 10-man Samp.
AC Milan head coach Massimiliano Allegri felt Milan "deserved more" than a loss.
"It was a balanced game and on the second goal we made a mistake in defence that allowed the counter-attack," he said.
"In any case, I am happy with the performance because we had several chances to equalize after Vucinic's goal.
"It's a shame about the result, especially Marco Storari's final save. We can improve, but also deserved more from this match," he said.
