Following a breath-taking end-to-end first half at the Ramon Sanchez Pizjuan Stadium that somehow ended goalless, both teams called on substitutes to make the difference with Dusan Vlahovic's opener cancelled out just six minutes later by Suso to send the semi-final tie into extra time.
Another substitute then grabbed the winner for Sevilla, with Erik Lamela heading home five minutes into extra time to set up a final showdown with Jose Mourinho's Roma in Budapest on May 31.
Jose Luis Mendilibar's side made a bright start to the contest and enjoyed a prolonged spell of possession in the final third, with Ivan Rakitic the main man dictating play and trying to find an opening for his teammates.
However, the first chance of the second leg came to Juve in the 14th minute when Nicolo Fagioli whipped a dangerous corner kick onto the head of unmarked defender Federico Gatti - who scored a last-gasp equaliser in the 1-1 first-leg draw last week - but his powerful header was beaten away by Sevilla shot-stopper Bono.
The game livened up and became stretched following that opening chance, and Sevilla came agonisingly close to breaking the deadlock shortly after when a diving header from Lucas Ocampos was brilliantly clawed off the line by Wojciech Szczesny diving to his near post.
Juve went up the other end moments later, with Angel di Maria running through on goal and dinking an effort over the crossbar after receiving a perfectly weighted through-ball from Adrien Rabiot.
Szczesny was called into action again to keep out a swerving speculative shot from Sevilla let-back Marcos Acuna from around 30 yards out, before the slightest of touches from Bono's right boot was enough to see a low-driven strike from Moise Kean skim the far post at the other end.
Juve were forced to make an early substitution as Fagioli was stretched off with an apparent rib injury following a collision with Nemanja Gudelj; Leandro Paredes came on as his replacement five minutes before the break.
The Italians thought they had opened the scoring when Manuel Locatelli ran in behind the Sevilla backline and scooped the ball back to Rabiot to smash the ball home, but Locatelli was caught fractionally offside.
Kean was arguably fortunate to not be given his marching orders as the Juve striker, who was booked just 10 minutes into the second leg for a late tackle, appeared to stamp on the calf of Ocampos, but the on-field referee decided against any further punishment.
Sevilla were then unfortunate to not be awarded a penalty in first-half stoppage time when Juan Cuadrado dived into a challenge and caught Oliver Torres' foot on the line of the penalty area, with a lengthy VAR review resulting in no spot kick given.
The second half began in frantic fashion, and Rabiot should have done better with an effort on the 54-minute mark, but the Frenchman could only drag his left-footed shot wide of the far post after being presented with acres of space to drive into Sevilla's penalty box.
With Juve lacking that clinical edge in the final third, Massimiliano Allegri decided to bring on Vlahovic and Federico Chiesa just after the hour mark, and the former made an instant impact just 60 seconds after coming off the bench.
The Serbian capitalised on the hosts' failure to clear their lines on the edge of the area, before delicately chipping the ball over the onrushing Bono to put the visitors ahead in the 65th minute.
However, just six minutes later, Sevilla restored parity with a substitute of their own as former AC Milan and Genoa attacker Suso fired a superb left-footed strike from outside the area into the top corner.
A curling free kick from Suso forced Szczesny to dive to his left and parry the ball away from danger in the 82nd minute, before the Polish shot-stopper pulled off a remarkable one-handed save in the 90th minute to deny Youssef En-Nesyri with a looping header.
Despite the best efforts of both teams to win the match in normal time, an additional 30 minutes was required and Juve started the brightest with two decent efforts from Vlahovic and Chiesa, but it was Sevilla who put themselves in front for the first time on the night five minutes into extra time when a fizzed delivery from Gil picked out substitute Lamela to head home from close range.
Acuna was sent off in the final five minutes after receiving his second yellow card for time wasting, and although the visitors threw bodies forward in search of a late equaliser, the 10 men of Sevilla held on to book their place in the Europa League final for the seventh time.
Juventus will hope to brush aside their disappointment when they return to Serie A action against Empoli next Monday, while Sevilla play host to Real Betis in La Liga on Sunday.