There were no goals in regulation time in the tight all-Segunda Division battles, with Deportivo only firing in goals late to avoid a shootout in Vitoria-Gasteiz.
Marco Sangalli netted in the 115th minute, tapping in on the goalline after goalkeeper Roberto Santamaria kept out an on-target shot but was unable to handle it.
Ranko Despotovic then sealed Deportivo's progression into the third round two minutes later, put through in acres of space before finishing with his right foot into the bottom left corner.
Earlier, Girona advanced past the second round with a 6-5 shootout win over Tenerife in Girona.
Uli Davila took Tenerife's second penalty but failed to convert, only for Girona's David Garcia to blow his chance to see the shoot-out to sudden death.
After Aday Benitez scored for Girona, Tenerife's Hugo Alvarez saw his spot-kick saved by Jorge Palatsi - the hero for the home side who stayed in the Spanish cup.
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