The hosts won the second-leg contest with a late own goal, but it was not enough as the visitors did just enough to advance in an emotional and exciting game
The visitors managed to sneak through thanks to a second-half goal by Roberto Soldado, winning on away goals after the tie finished 2-2 on aggregate.
After opening the scoring, Los Che saw the scoreline reversed as Ivan Rakitic brought things level before an own goal from Victor Ruiz in the 91st minute added late tension.
With a 1-0 advantage after Jonas' goal in the opening leg, it did not take Valencia long to suffer.
Sevilla earned all of the early momentum, bombarding Diego Alves from all angles during the first 35 minutes, but the Brazilian stood strong, saving Unai Emery's team from an early deficit with athletic and instinctive saves.
It was a tough opening half an hour for Valencia, as their out-of-sync defence struggled to handle the pressure and speed of their opponents, looking a step slow to each ball.
For the final 15 minutes of the opening half, Valencia managed to turn the tide, gaining control and producing a flurry of their own chances, mainly through Jonas, but Javi Varas was able to match the rival custodian, preserving the goalless scoreline entering the break.
In the second half, the offensive display was reduced to possession by the visitors and occasional chances by the Rojiblancos, until the crucial opener.
With Spain coach Vicente del Bosque watching from the stands, Soldado was sent through by a low ball near midfield from Jonas. The striker slipped past the defenders before finishing calmly into the right side of the net.
Moments later, the hosts restored parity with Rakitic putting the ball away from close after Alves saved a long effort from Martin Caceres but failed to fully hang onto it.
Sevilla looked to climb back and took the lead thanks to Ruiz, who chested the ball into his own net late off a free kick, making the visitors night even more hectic. But, needing three goals, the home side could not find another one in added time.
Emery's men were able to hold on and will now face Levante for a spot in the semi-finals after the club downed Alcorcon earlier on Wednesday.