The visitors completed a turnaround by overturning the two-goal first-leg deficit and eventually securing victory in a dramatic penalty shootout
The visitors won 2-0 on the night to overturn Betis' first-leg lead and then held their nerve in the penalty shoot-out, winning 4-3 despite Vitolo missing their first kick.
Goals from Jose Antonio Reyes and Carlos Bacca either side of the break on Thursday inflicted Betis' first home defeat of the season in Europe and it was Sevilla which prevailed from 12 yards, as Alfred N'Diaye and Nono let Vitolo off the hook.
The result marked the first time Sevilla has overturned a first-leg home defeat in UEFA competition, leaving the hosts to focus on their battle for La Liga survival.
Having lost the first leg to goals from Leo Baptistao and Salva Sevilla, Unai Emery named a bold starting XI at Benito Villamarin, with Bacca and Kevin Gameiro starting together in attack.
Sevilla attacked relentlessly in the opening stages and took a deserved lead 20 minutes in as Reyes slotted Alberto Moreno's left-wing delivery into the bottom-left corner.
Gabriel Calderon's Betis struggled to create any kind of momentum but will count itself unfortunate not to have been awarded a penalty just after the half-hour mark, with the ball appearing to strike Diogo Figueiras on the arm.
After a nervous start to the second half, Betis started to show more attacking intent as Ruben Castro headed wide 55 minutes in, before Baptistao struck just wide from 25 yards less than a minute later.
Sevilla goalkeeper Beto kept his side in with a shout thanks to a crucial stop in the 64th minute, parrying Baptistao's half-volley after a clever chipped pass by Nono.
The away side drew level 15 minutes from time. Moreno exploited space down the left before drilling a low cross into the danger area for Bacca to knock past Adan unchallenged.
With the score level after 90 minutes, the match entered extra-time and Sevilla should have clinched the tie.
Adan pulled off an excellent stop when one-on-one with Gameiro, before palming away Bacca's headed effort on the rebound.
The Betis keeper saved from Vitolo to give the hosts the early edge in the shoot-out, but N'Diaye struck the post and Nono dragged wide to give Sevilla a place in the quarterfinals.