Iran Press/ Europe: Dusan Vlahovic had put Juventus ahead, but Suso leveled the tie with a brilliant left-footed strike.
The decisive goal came five minutes into extra time when Bryan Gil crossed for Erik Lamela to head home.
Sevilla defender Marcos Acuna was sent off late on but the hosts held out and will face Roma in the final.
After a steady opening, the game sprang into life midway through the first half with both goalkeepers called into action.
Sevilla's Yassine Bounou kept out Federico Gatti's header at one end before Wojciech Szczesny superbly clawed Lucas Ocampos' diving header away at the other.
Buonou then got the slightest of touches to tip Moise Kean's goalbound effort onto the far post after the Juventus striker's smart turn in the box and the half ended with Sevilla dismayed not to be awarded a penalty when Juan Cuadrado caught Oliver Torres with a flying challenge.
The chances kept coming in the second half as Juve twice went close - Adrien Rabiot shooting wide after bursting clear before Bremer headed just past the post from Leandro Paredes' cross.
Juventus took the lead on 65 minutes as Vlahovic brushed past a defender and finished with a clever dink just 58 seconds after coming on.
Sevilla equalized six minutes later through their substitute. Suso collected Lamela's pass, jinked away from his marker, and lashed the ball into the corner from 20 yards.
The hosts nearly won it in the last minute of normal time but Szczesny leaped to tip Youssef En-Nesyri's header over - although the Juve keeper could do nothing to deny Lamela in extra time.
The visitors responded well as they hunted a goal to send the tie to penalties but Federico Chiesa blazed their best chance of extra time over the bar.
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