Juventus' players pose with the trophy after winning the Italian Tim Cup final football match AC Milan vs Juventus on May 21, 2016 at the Olympic Stadium in Rome. Juventus won 0-1 in the extra time. PHOTO: AFP
It means Juventus become the first Italian team to win successive league and Cup doubles, a year after ending a 20-year wait to achieve the feat.
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AC Milan's defeat means the seven-time European champions failed to qualify for Europe next season. Sixth place finishers Sassuolo will be handed a Europa League place.
Milan put everything into a match most expected to be a one-sided in Juve's favour but came away lamenting their failure to convert a host of chances.
PHOTO: AFPMorata came off the bench to replace Hernanes in the second period of extra time, and only two minutes later the Spain international fired a superb first-time strike from Juan Cuadrado's chipped pass past a well beaten Ganluigi Donnarumma to wrap up the double for Massimiliano Allegri's men.
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Morata, who according to reports is attracting interest from Premier League clubs, told Rai Sport: "I'm really happy for the Cup, it's another one for Juve's cabinet.
"Yesterday my friends asked me if I would play, and I said no. So they [the team] told me to go on and score. If only all finals were like this... it's the beautiful part of football.
PHOTO: AFP"My father's in the stands and my mum's watching from home."
But on all-round performance, the champions were barely worthy winners after a comparatively stirring Milan performance given the disastrous season of Cristian Brocchi's men.
Juve midfielder Paul Pogba had claims for a penalty waved away in the early stages when he was upended by Davide Calabria.
But from then on, it was mostly all Milan.
Calabria overcame his early jitters to whip in an inviting cross for Giacomo Bonaventura on 13 minutes but the midfielder lashed over from yards out.
It was the first of several misses in a one-sided opening half that saw Matteo De Sciglio, Andrea Poli and again Bonaventura come close to breaking the deadlock.
PHOTO: AFPMilan continued apace after the break, Neto forced down low to stop De Sciglio's low strike in the opening minutes, although a Juve counter had Donnarumma out his net in quick fashion to palm away Mario Lemina's rasping cross away from the feet of Dybala as the Argentine teed up to strike.
Juve enjoyed a brief spell of promise, but when Pogba muscled his way into the area his deflected strike was met by the right hand of an alert Donnarumma at the near post.
Stephan Lichsteiner then fired a header into the hands of Donnarumma from Pogba's cross, and when Mario Lemina found Mario Mandzukic with a great through ball moments later Donnarumma parried the Croatian striker's half-volley on the turn on 74 minutes.
Allegri replaced Lichsteiner with Cuadrado a minute later, and six minutes from the finish M'Baye Niang replaced Poli.
Into extra-time, Juve got another sniff when a tiring Riccardo Montolivo gifted the ball away in midfield and Mandzukic set up Pogba for a piercing strike from distance that Donnarumma did well to parry.
Carlos Bacca then fired a spectacular overhead kick just over and minutes later the hosts had claims for a penalty waved away when Honda came down after clashing with Giorgio Chiellini.
Milan were tiring and Juve's late changes made the difference when a counter saw Cuadrado allowed to fire a chip over for the unmarked Morata, who swept past Donnarumma with ease.
Milan threatened a late leveller when Jose Mauri's low drive from the edge of the area shaved the post, but it was not to be for the struggling Serie A giants
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