Nuno's men had been put under pressure in the race for the top four in La Liga by Sevilla, who moved level on points with Valencia by beating Levante on Tuesday.
Rodrigo De Paul put Valencia ahead after 61 minutes with a smart finish in the bottom-right corner, but the game turned when Nicolas Otamendi was harshly sent off in the 82nd minute following a collision with Bilbao's Mikel San Jose.
And Valencia's indignation increased in the 90th minute when Aritz Aduriz tapped home the leveller for Bilbao from what looked to be an offside position, after Borja Viguera's deflected effort struck the post.
Elsewhere, Felipe Caicedo scored twice as Espanyol won 3-0 at Villarreal and Elche remain just three points clear of safety following a 1-0 home defeat to Getafe.
Fredy Hinestroza netted the winner for Getafe six minutes from time with a near-post header from a corner.
