The champions made a bright start and went in front on 10 minutes at Anoeta through Mario Mandzukic, only for Carlos Vela to level soon after.
However, a close contest was turned on its head when Guilherme Siqueira was sent off for a second bookable offence early on in the second half.
Atletico's hopes of a sixth straight win in all competitions faded and they were denied even a share of the spoils when Imanol Agirretxe headed home an 82nd-minute winner.
"It has not been a good day for us," said Moya. "Now we have to lift our heads up and bounce back."
Substitute Cristian Ansaldi echoed those thoughts when insisting Atletico had nothing to be ashamed of.
He added: "There is no need for us to lower our sights. We have to continue working for each other and with humility."
Atletico showed signs of tiredness as the game wore on, having had to travel to Sweden for a 2-0 UEFA Champions League win over Malmo in midweek.
However, Diego Simeone refused to be too downbeat, commenting: "The game had great intensity and was constantly swinging from one way to the other."