Teenager Sergio Canales netted a superb free-kick as Racing Santander also advanced after a 3-0 victory at Osasuna that sent the Cantabrians through 5-1 over the two legs.
Sevilla, winners in 2007 and conquerors of holders Barcelona in the last round, will play Getafe or Real Mallorca in next month's semi-finals, while Racing face Atletico Madrid or second division Celta Vigo.
Striker Rodolfo Bodipo put Deportivo ahead shortly before halftime at Sevilla's half-empty Sanchez Pizjuan stadium when he slid in to turn the ball high into the net and raise the visitors' hopes of a comeback.
Celta, the only side left from outside the top flight, host Atletico on Thursday after last week's first leg in Madrid finished 1-1 and Mallorca are looking to overturn a 2-1 deficit when they visit Getafe. The final is in May.
