Los Che, who led 2-1 in the tie from the first leg at the Mestalla, found themselves a goal down in the 73rd minute.
With just four minutes left in the game, Alaves levelled the tie through Ruben Sobrino, who drilled home from close range following a free kick into the area, to force extra time.
The additional 30 minutes of football could not separate the two sides either, ensuring that the tie would have to be settled via a penalty shootout.
Sobrino then blasted his effort high above the bar in the final penalty kick to ensure that Marcelino Garcia Toral's side would progress to the final four of the competition.
ALAVES (4-4-2): Sivera; Aguirregabiria, Laguardia, Ely (Demirovic 85'), Duarte; Gomez (El Haddadi 72'), Pina, Torres, Pedraza; Guidetti (Perez 72'), Sobrino
VALENCIA (4-4-2): Jaume; Montoya, Gabriel (Garay 23'), Vezo, Gaya; Maksimovic, Parejo, Kondogbia, Guedes; Vietto (Rodrigo 59'), Zaza