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.
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.
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
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.
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.
