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.
Valencia survived a torrid final few minutes when Bilbao's David Lopez missed a late penalty.
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.
"It sets us up nicely (for the visit of Real next week). It's always an attractive game and even more so in the situation we are now in," Emery said.
Champions Barcelona are top with 36 points, having played a game more than their rivals, after beating Deportivo La 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.
Mallorca's problems off the field, where attempts to take over the financially troubled club have ended in acrimonious disputes, have not affected the team's form inside the Ono Estadi this season.
They have won seven from seven there under the wily leadership of coach Gregorio Manzano, and in striker Aduriz they have a proven goalscorer who has now scored seven this year.
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.
CAPDEVILA DOUBLE
Spain left back Joan Capdevila scored a late winner for nine-man Villarreal, his second of the game, to secure a 3-2 comeback win at home to Getafe.
Villarreal had midfielder Cani sent off just after the re-start when he conceded a penalty which Roberto Soldado netted to give the visitors a 2-1 lead.
Capdevila's international team mate Santi Cazorla returned from injury, coming off the bench to net a magnificent volley for the home side's equaliser in the 67th minute. Villarreal's Angel Lopez was sent off at the end.
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.
Their 18-year-old striker Sergio Canales scored twice helping them to only their second win this season.
For Espanyol it was their fourth defeat in a row without scoring, and they have now conceded eight in their last two outings.
Racing climbed to 18th with 10 points, ahead of Malaga, who drew 1-1 with Osasuna, and bottom
