Bilbao's 16-year-old striker Iker Muniain came off the bench to give the home side a 58th-minute lead, but three minutes later Spain's David Villa volleyed Valencia level, with his 11th of the campaign.
Bilbao had defender Koikili sent off in the 73rd and Valencia profited when French defender Mathieu scored with a rasping drive nine minutes later.
Unai Emery's men moved up to third with 28 points one ahead of Sevilla who drew 1-1 at home to ten-man Real Valladolid on Saturday, and three behind second-placed Real Madrid.
Champions Barcelona are top with 36 points, having played a game more than their rivals, after beating Deportivo Coruna 3-1 away. Real Madrid are five points behind after Saturday's 4-2 victory at home to Almeria.
Aritz Aduriz scored twice as Real Mallorca hammered Real Zaragoza 4-1 to maintain their perfect home record, putting them sixth with 25 points, one behind Depor.
The powerful frontman set them on their way with goals either side of the break, and Mario Suarez and Alhassane Keita made it 4-0 before Ander Herrera pulled one back for the visitors at the end.
Spain left back Joan Capdevila scored a late winner for nine-man Villarreal to secure a 3-2 comeback win at home to Getafe. Villarreal are 11th with 15 points, three behing Getafe in ninth.
Racing Santander's new coach Miguel Angel Portugal earned his first win since taking the helm last month, the Cantabrians sweeping aside Espanyol 4-0 away.
Racing climbed to 18th with 10 points, ahead of Malaga, who drew 1-1 with Osasuna, and bottom club Xerez.
Defender Marco Cassetti slammed in 11 minutes from time to give AS Roma a 1-0 win over Lazio on Sunday in a derby held up by fans throwing firecrackers.
Roma's David Pizarro was dismissed for two bookings late on while there was a customary scuffle between players at the end. In-form Roma are now just a point off the Champions League places after a bad start while Lazio coach Davide Ballardini, who continually overlooks Goran Pandev after their top striker asked for a close-season move, is clinging to his job.
Jonathan Biabiany scored his first two Serie A goals to put Parma clear in fourth place after a 2-2 draw at Genoa.
Promoted Parma are flourishing back in the top flight and lie a point ahead of Genoa, Sampdoria, Roma and Fiorentina, who beat Atalanta 2-0 thanks to Juan Vargas and Alberto Gilardino. Third-placed Juventus, five above Parma, overcame leaders Inter 2-1 on Saturday when AC Milan, in second, beat Samp 3-0.