With penalties looming at the San Siro Sanchez poked home from close range in the 121st minute of a gruelling contest to hand Simone Inzaghi his first trophy since arriving at Inter to replace Antonio Conte in the summer.
Chile forward Sanchez, who has started just four matches this season, was submerged by celebrating teammates as he charged to the sidelines to celebrate his winner, which came after Lautaro Martinez's first-half penalty had levelled Weston McKennie's opener for Juve.
"That's what great players are like, the more we play, the better we feel," Sanchez told Mediaset.
"I feel like a caged lion, if they let me play I'm a beast."
The win continues Inter's sparkling form, with eight wins from their last eight matches in Serie A, which they lead by a point from city rivals AC Milan.
For Juve meanwhile it was another disappointment in a season which had looked like it was getting on track in recent months after an inconsistent opening.
Martinez could have put Inter ahead within eight minutes when, completely unmarked, he somehow didn't meet Stefan De Vrij's header from a corner, and seconds later flashed a first-time effort wide when laid on by Nicolo Barella.
McKennie silenced the Inter fans in the 25th minute when after Alvaro Morata collected on the edge of the area the Spain forward clipped over a cross which was deflected straight into his American teammate's path.
However Inter were level 10 minutes later thanks to Martinez, who rifled home a precision spot-kick after Edin Dzeko was taken out by Mattia De Sciglio.
Juve started the second half with their tails up but apart from two efforts from Federico Bernardeschi it was Inter who were the dominant team after the break.
They had the best chance to break the deadlock in normal time when Mattia Perin pushed Denzel Dumfries' close-range header onto the bar just before the hour mark but struggled to create much more danger despite being clearly the better team.
Inter went close again five minutes into extra-time and again it was Martinez who should have done better, this time heading narrowly wide from Hakan Calhanoglu's corner.
However with Juve preparing to send on Leonardo Bonucci ahead of spot-kicks, Alex Sandro miscontrolled a cross, allowing Matteo Darmian to nip in and push the ball to Sanchez, who sent the Inter fans into raptures with his simple finish.