The visitors won 2-0 on the night to overturn Betis' first-leg lead and then held their nerve in the penalty shoot-out, 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 who prevailed from 12 yards, as Alfred N'Diaye and Nono let Vitolo off the hook.
The result marked the first time Sevilla have 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.
With Betis sitting on the first-leg lead, the visitors were invited forward right from the start and they went close to finding the net in the 11th minute, but Gameiro narrowly failed to reach Bacca's low cross.
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 themselves 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.
Betis' aggregate lead was almost wiped out three minutes into the second period as Stephane M'Bia played a one-two with Gameiro before shooting straight at Adan from point-blank range.
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.
And the away side drew level on aggregate 15 minutes from time as 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 quarter-finals.
