The Montenegro striker hit an unstoppable right-foot drive past Milan goalkeeper Marco Amelia in the first half of extra-time to ensure the game finished 2-2.
Juventus led the Serie A leaders 2-1 after the first leg in February and increased their advantage when the 37-year-old Alessandro Del Piero scored from close range in the first half.
The visitors, who had overtaken their opponents in Serie A since the first leg took place, came back into the tie when left back Djamel Mesbah equalised with a header early in the second half.
Vucinic missed a golden opportunity to put the match to bed when he shot into the side netting before Milan's Argentine striker Maxi Lopez turned on the edge of the area and blasted the ball into the top corner from 20 metres to force extra time.
Juventus will play the winners of Wednesday's second semi-final between Napoli and Siena.
Milan had looked the more threatening in a feisty and up-tempo first half.
Zlatan Ibrahimovic forced a smart save from Juventus keeper Marco Storari on 21 minutes after cutting into the area from wide on the left before Clarence Seedorf sent a swerving shot from distance just wide.
The home crowd reacted angrily as heavy challenges from Sulley Muntari and Alberto Aquilani resulted in yellow rather than red cards but their mood was lightened when the deadlock was broken on 28 minutes.
Stephan Lichtsteiner marauded forward on the right before crossing for Del Piero to flick the ball over the on-rushing keeper and tuck it into the empty net from close range.
The lead lasted until five minutes into the second half when Mesbah was allowed to ghost into the Juventus box unchecked and head a long ball forward from Philippe Mexes into the net.
Vucinic ignored the pleas of better-placed team-mates before missing the target and his profligacy was punished when Lopez, a second-half substitute for Ibrahimovic, turned sharply on the edge of the area and let fly with an unstoppable drive from 20 metres.
Juventus began extra time at a breakneck pace and should have scored when Vucinic failed to round the Milan keeper when clean through on goal.
He made amends, however, when he picked out the top corner of the Milan net from 25 metres with a nonchalant effort made with little backlift.
