No Sevilla side had ever won a European tie after losing the first leg at home, but Unai Emery's men levelled the scores up on aggregate thanks to goals in either half from Jose Antonio Reyes and Carlos Bacca at the Benito Villamarin.
That sent the game to extra-time and then penalties, where Sevilla won 4-3 despite seeing their first penalty from Vitolo saved by Antonio Adan.
Betis looked to be on their way through to the quarter-finals after that miss as Ruben Castra, Salva Sevilla and Antonio Amaya all emphatically crashed home unstoppable penalties.
However, Sevilla kept in touch, and Betis lost their advantage when Alfred N'Diaye's effort came back off the post.
Sevilla midfielder Ivan Rakitic then slotted home to put his side 4-3 up, heaping the pressure on Betis substitute Jose Nono, who subsequently produced the worst kick of the shoot-out as his soft effort was saved by Antonio Beto to give the visitors victory.
After the first-leg upset, Sevilla would have been desperate for an early goal to both give themselves a lifeline and also try to deflate the home crowd, and they got the breakthrough they were looking for inside the opening 20 minutes.
The Betis defence failed to cut out Alberto Moreno's cross the left and the ball fell to Reyes near the penalty spot, where the former Real Madrid winger swept a first-time shot past Adan.
That was almost the ideal start for the 2006 and 2007 UEFA Cup winners, but Betis steeled themselves after that setback and the visitors found it hard to create much else in the first half.
In fact, they had to wait until the 49th minute for their next decent opening, when Stephane Mbia very nearly found the back of the net only to be denied by the feet of Adan.
Betis responded to that scare by creating two good chances of their own, both of which were missed by first-leg goalscorer Leo Baptistao.
His first effort was a hopelessly mis-hit strike from the edge of the area, but the second was much cleaner and it needed the outstretched foot of Beto to keep it from drawing Betis level on the night.
A goal then would probably have ended the tie but Sevilla survived and, with 15 minutes to go, got the goal that levelled things up on aggregate.
Highly-rated full-back Moreno was again involved with a superb low cross from the left that was inch-perfect for Bacca to sidefoot home from eight yards out.
Bacca's strike sent the match to extra-time, despite some late Betis pressure, and the additional 30 minutes could not separate the two rivals either.
That was largely because of a brilliant double save from Adan in the 104th minute to keep out Kevin Gameiro's shot and Bacca's header from the rebound.
Adan then gave his side a flying start in the penalty shoot-out when he saved Vitolo's opening effort, but it was his Sevilla counterpart Beto who was celebrating at the end as he pushed wide Nono's spot-kick.