Rayo has lost two straight after its first-round victory.
Elsewhere, Elche forward Yiadom Boakye also netted deep in stoppage time to snatch a 2-2 draw at 10-man Almeria in a clash of promoted sides.
Almeria dominated early at its Juegos Mediterraneos Stadium with forward Fernando Soriano heading home Rafael "Rafita" Ramos' cross at the back post eight minutes in.
But Elche's Eduardo Albacar made it 1-1 in the 32nd by scoring a penalty after Sebastian Dubarbier handled the ball in the area.
Almeria went back ahead in first-half stoppage time when Elche defender Ionut Sapunaru was also called for a handball in the box, giving Jose "Verza" Garcia the opportunity to fire in the spot kick.
The match turned in Elche's favor in the 67th when Dubarbier was dismissed with his second booking. Almeria held on until five minutes into added time when Boakye spun in the area and drove the ball in.
Barcelona visits Valencia while Real Madrid hosts Athletic Bilbao on Sunday, when Real Sociedad welcomes Atletico Madrid, Malaga plays at Sevilla, and Espanyol faces Real Betis.
On Saturday, Villarreal visits Osasuna, Valladolid hosts Getafe, and Celta Vigo plays Granada.