Inter are still hoping for this first domestic treble in more than a decade, but to retain their title they need AC Milan to drop at least three points from their final two matches.
The defeat in the Derby d’Italia ended record 14-time winners Juventus’ hopes of a trophy this season – the first time that has happened since 2010-11.
Inter got off to a great start when Nicole Barella curled a sublime right-foot shot into the top corner after cutting inside from the left to score the fastest Coppa Italia final goal since 2015.
The goal seem to energize Juventus and Inter goalkeeper Samir Handanovic had a make a series of fine saves, the best one denying Dusan Vlahovic and Paulo Dybala shot wide as Juventus chased an equaliser before the break.
Massimiliano Allegri must have given a good half-time talk as his side were ahead within seven minutes of the restart thanks to goals from Sandro and Vlahovic, who followed up his own saved effort after a quick counter-attack.
A disputed penalty got Inter back on level terms as Calhanoglu held his nerve to force extra time after Lautaro Martinez was judged to have been brought down by Leonardo Bonucci.
Inter added a second penalty in extra time when Matthijs de Ligt fouled Stefan de Vrij and Ivan Perisic then stepped up from the penalty spot to put Inter back in front after 99 minutes.
Perisic added a stunning fourth just three minutes later, taking a touch before firing into the top corner from the edge of the box to put the contest out of Juventus’ reach.
Afterwards Inter boss Simone Inzaghi said that he was so proud and impressed with Inter’s ability to come back from trailing situations:
“It was a good performance against a very, very strong opponent. We did well in the first half-hour, slightly lost our shape towards the break and then had a bad approach to the second half, conceding two really avoidable goals. We should’ve been more careful,” Inzaghi told Sport Mediaset.
“However, this team never gives up. If I ever needed a response, I got nothing but responses from this side from July to today, it’s been one long whirlwind of emotions.
“Now we have two more Serie A matches to play and we want to end the season as well as possible. It is also only fair we enjoy these moments, it had been a decade since we won the Coppa Italia or the Supercoppa, so I am happy for the club, these marvellous lads and the fans who stuck by us from start to finish.”