Morata blasted home Juan Cuadrado's cross with his first meaningful touch after joining the Stadio Olimpico fray in the second period of extra time.
Listless attack from both sides led to an uninspiring stalemate after 90 minutes - and just when both teams were gearing up for penalties, Spain hitman Morata settled the argument.
Juventus became the first team in Italy's history to pull off back-to-back Serie A and Coppa Italia doubles, denying AC Milan European qualification in the process.
Milan's seventh-place Serie A finish and failure to land the Coppa Italia means Sassuolo have now qualified for Europe for the first time.
Max Allegri's Juventus side won Serie A by nine points, but struggled to impose any authority on a low-quality cup final in Rome on Saturday night.
The two shot-shy sides laboured through the initial 90 minutes failing to create any fully-fledged chances for goals.
Juve were furious to be denied a penalty when Davide Calabria appeared to clip Paul Pogba in the box in the opening exchanges however. But after that both sides flattered to deceive and almost seemed to tiptoe around each other's final third.
Giacomo Bonaventura fired over from distance for Milan, before flicking a low cross wide from mere yards out, while Andrea Poli side-footed meekly over despite acres of time and space.
Lacklustre Juve just about upped the pace after the break, with Milan keeper Gianluigi Donnarumma palming a cross away from Mario Mandzukic.
Pogba's low cross deflected off Cristian Zapata as time ticked away, but again Donnarumma was alert to the danger, tipping round the post.
Pogba drew a solid save from Donnarumma in the first period of extra time, with a 20-yard drive, before Carlos Bacca flicked an overhead kick too high for Milan.
Former Real Madrid striker Morata showed both sides' attacking talents just how to finish when he finally joined the fray however.
Pulling away from the cover in the box, the 23-year-old took Juan Cuadrado's cross on the half-volley, and lashed the ball past the despairing Donnarumma.