Goals from Artiz Aduriz and Raul Garcia secured victory for the visitors on the night, but Benat's penalty miss in the shootout allowed Kevin Gameiro to secure a last-four berth.
Sevilla's quest for a third successive Europa League title remains on course, but only just, after they needed penalties to see off Athletic Club to reach the semi-finals.
The hosts had lost just one of their last 18 matches at the Ramon Sanchez Pizjuan prior to Thursday's quarter-final second leg, but trailed 2-1 after 90 minutes to leave the tie locked at 3-3 on aggregate.
After falling behind to a 57th-minute strike from tournament top scorer Aritz Aduriz, Sevilla hit back two minutes later through Kevin Gameiro.
Athletic's attempts to force extra time were hampered by an injury to Aduriz, but Raul Garcia popped up to head the crucial goal 10 minutes from time.
Neither side found a winner during the additional 30 minutes and, after Benat sent his spot-kick straight into the arms of David Soria, Gameiro - who had earlier gone down with a calf problem - stepped up to convert the decisive penalty in a shoot-out that ended 5-4 to the holders.
Early penalty claims went unrewarded when Sevilla felt a Mariano cross struck Benat on the arm, before Aduriz knocked the ball around Soria at the other end but saw his effort from a tight angle touched behind by Timothee Kolodziejczak.
For all the attacking intent, particularly from Athletic, who were without the injured Inaki Williams, neither side was able to find a breakthrough in the opening period, although Garcia forced Soria into a diving save before the break.
Sevilla started the second half on the front foot and Iago Herrerin had to be alert to push wide a rifled Vicente Iborra strike from the edge of the box, before Mariano half-volleyed over.
But it was Athletic who eventually broke the deadlock when Aduriz created an angle for the shot and sent an effort squirming past Soria, who will do some squirming of his own when he watches the replay.
Sevilla responded almost immediately, though, with Gameiro tapping home from a Grzegorz Krychowiak ball into the six-yard box.
Mikel San Jose – booked for dissent moments earlier – was perhaps lucky to remain on the field when, as the last man, he clumsily dispossessed an advancing Gameiro in the 69th minute, but Athletic were soon left reeling by the departure of Aduriz through injury.
Gameiro proved an increasing nuisance for the Athletic defence, seeing another powerful strike kept out by Herrerin before Steven N'Zonzi headed wide from a promising position.
But Garcia then netted Athletic's crucial second after Gameiro had smashed yet another effort over at the other end. The former Atletico Madrid man got on the end of a lofted Benat cross to send a looping header over Soria, who once again failed to cover himself in glory as he watched the ball float beyond him.
With the game forced to extra time, Markel Susaeta was left with his head in his hands after sending the ball bouncing wide from a one-on-one with Soria, before a Yevhen Konoplyanka free-kick found its way onto the right-hand upright via the head of a defender.
Tempers flared as extra-time reached the midway point, but Herrerin kept his focus to get down to his left and keep out a Konoplyanka set-piece.
With neither team able to put the match to bed, it was left to penalties to separate the sides, and Gameiro, whose turn was held up by a scuffle in the centre circle after Oscar De Marcos' efforts to put off the Sevilla striker, kept hopes of a remarkable treble alive for Unai Emery's men.
