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."
He added: "There is no need for us to lower our sights. We have to continue working for each other and with humility."
However, Diego Simeone refused to be too downbeat, commenting: "The game had great intensity and was constantly swinging from one way to the other."
