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.
However, the home side controlled the match in the second half and striker Luis Fabiano and defender Lolo had efforts ruled out for offside.
Racing midfielder Canales, 18, earlier showed why Real Madrid are hoping to sign him next week when he sent a curling left-foot strike past Osasuna goalkeeper Roberto Fernandez from the corner of the area with just over 10 minutes left.
Striker Xisco had put Racing ahead at the Reyno de Navarra stadium in Pamplona when he battled into the penalty area in the sixth minute and coolly poked the ball into the bottom corner.
Defender Henrique doubled the lead in the 23rd, rising to head home from a corner, before Canales added a third.
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.
