Nicolo Barella's sensational 10th-minute opener appeared to have put Inter on course to repeat its Supercoppa Italiana triumph over Juve in January.
But two goals in the space of just over two second-half minutes from Alex Sandro, aided by Alvaro Morata, and Dusan Vlahovic turned the tide in Juve's favour.
There was a twist in the tale late in normal time, though, Hakan Calhanoglu netting from the spot to send the game to an extra half-hour, in which Perisic also converted from 12 yards and then sealed victory with a fine half-volley.
Barella broke the deadlock in stunning fashion as he cut in from the left and unleashed an unstoppable long-range strike into the top-right corner.
Juve's response was impressive, Samir Handanovic forced into action to deny Paulo Dybala and then producing a superb save to turn Vlahovic's near-post effort behind.
Handanovic tipped Matthijs de Ligt's header over the crossbar on the half-hour mark, with Dybala then firing narrowly wide of the left-hand post.
But Juve's persistence was rewarded five minutes after the restart as Handanovic gifted them an equaliser.
Morata seemingly got a slight deflection on Alex Sandro's drive from the edge of the area, which Handanovic allowed to squirm through.
Juve's comeback was completed two minutes later as Inter were caught cold by a rapid counter.
Dybala threaded a perfectly weighted pass through to Vlahovic, who showed the composure to drop his shoulder and unseat the chasing Danilo D'Ambrosio before hitting home on the rebound after his initial effort hit Handanovic in the face.
Matteo Darmian drew a save from Mattia Perin but Inter did not look like forcing extra-time until Leonardo Bonucci was adjudged to have brought down Lautaro Martinez in the 78th minute.
Calhanoglu emphatically levelled, sending an excellent penalty in the top-right corner, and Perisic found the other corner with his spot-kick nine minutes into extra time after a VAR review into De Ligt's challenge on countryman Stefan de Vrij.
Perisic gave Inter breathing room in style, rendering Perin helpless with a sumptuous half-volley to cap an absorbing final.