The hosts won the contest with a late own goal, but it was not enough as a goal by the visitors after an hour was just enough to advance in an emotional, exciting contest
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, Valencia saw the scoreline reversed as Ivan Rakitic brought things level before an own goal from Victor Ruiz in the 91th minute added late tension.
With 1-0 advantage after Jonas' goal in the opening leg, it did not take Los Che 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 Valencia from an early deficit with athletic, instinctive saves.
It was a tough opening half an hour for Unai Emery's side, as his out-of-sync defense struggled to handle the pressure and speed of their opponents, looking a step slow to each ball, having to wait most of the first half for their first chance.
For the final 15 minutes of the opening half, Valencia managed to turn the tide, gaining control and producing a flurry of its 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 Valencia and occasional chances by Sevilla, 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 for his second goal away from home on the season.
Moments later, the hosts got one back 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, making the visitors' night even more hectic. But, needing three goals, the home side could not find another late one.
Emery's men were able to hold on and will now face Levante for a spot in the semifinals after the club downed Alcorcon earlier on Wednesday.