The narrow triumph has secured Juve's place in history as the first side in Italian football to win a domestic league and cup double in consecutive seasons, and they have now won the trophy on 11 occasions.
AC Milan began the better of the two sides in Rome, with some impressive energy, strong running and pressure from Cristian Brocchi's men.
Nineteen-year-old winger Davide Calabria darted down the right flank and did superbly well to pick out Bonaventura at the back post after 15 minutes. However, the former Atalanta midfielder controlled well but could only strike his shot wide of goal.
Former Manchester United left-back Patrice Evra provided Juve with their best chance of the match on the 30-minute mark, driving an effort on goal that was deflected into the side netting. That move appeared to spring life into Allegri's men, as they began to play with increased confidence and intent in their opposition's half.
Paulo Dybala showed his ambition with a fine run down the right-hand side before feeding Stefan Lichtsteiner down the byline. The Swiss international's low cross found Mario Mandzukic, but the Croat could only bundle his finish, with goalkeeper Gianluigi Donnarumma flicking his effort to safety.
Juve's impressive late showing in the first 45 minutes continued following the half-time interval. Paul Pogba—who was restrained during the opening half—tested Donnarumma with a pair of free-kicks, but neither was close enough to the target to truly test the youngster between the sticks.
Extra time appeared inevitable as the clock began to tick down in the Italian capital, with Juve turning to desperation in a bid to seal victory in normal time. Ex-Manchester United playmaker Pogba struck a fizzing effort from 30 yards out that flew off target before Mario Lemina and Dybala struck similar wayward efforts soon after.
The quality of the contest reduced in extra time, as neither side was able to create significant chances on goal, with both sets of players tiring late on in the match.
However, with 10 minutes remaining in the second period of additional time, Morata unleashed a moment of brilliance to net the winner. Arriving on the scene just two minutes earlier in place of Brazilian midfielder Hernanes, the former Real Madrid striker produced a sublime half-volley to finally beat Donnarumma.
Cuadrado's scooped effort towards the back post was perfectly met by the Spaniard, whose controlled finish secured an unprecedented second successive league and cup double for Juventus.