Cadiz's elimination on Thursday came a day after Almeria fell 1-0 to fourth-division club Barbastro.
Jorge Pesca scored a 65th-minute winner for Arandina. The club based in Aranda de Duero, a small city in northern Spain, had never advanced past the second round.
It was the 13th game in a row without a win for Cadiz, which sits in 16th place in the Spanish league, three points from the relegation zone. It advanced in the first round of the Copa with a win on penalties against fifth-division club Badalona.
First-division club Granada was eliminated after being disqualified for fielding an ineligible player in the first round.
Last-place Almeria lost Wednesday to remain with only one victory in 17 matches in all competitions this season — against third-division club Talavera in the previous Copa round. Cadiz has only two wins in all competitions this season, and none since September.
Three top-tier clubs advanced past lower-division teams on Thursday — Girona defeated Orihuela 5-2, Athletic Bilbao beat Cayon 3-0 and Celta Vigo got past Sestao River 2-1.