Valencia stayed third in the table with 47 points, but city rivals
Zaragoza, who finished the match with nine men, humbled high-flying Valencia 2-1 despite falling behind to a Pablo Hernandez goal after only nine minutes.
They had defender Pablo Alvarez sent off in the 21st minute but Apono levelled from the penalty spot at the half hour mark.
The midfielder bagged the winner in the 76th and Zaragoza had midfielder Franco Zuculini sent off near the end of the match.
The home side left to a barrage of whistles as the pressure mounted on their coach Unai Emery.
Colombia striker Falcao pounced on a loose ball to grab the opener for Atletico just after halftime at the Calderon, and stooped low to head in the second in the 70th.
Javi Martinez pulled one back for Bilbao at the end, but the side who knocked Manchester United out of the Europa League last week, slipped to their third successive league defeat.
Atletico were seventh with 39 points, ahead of Bilbao in ninth with 37.