An Aritz Aduriz double either side of a Roberto Soldado penalty put Bilbao ahead at the break at Valencia's Mestalla arena.
After Herrera's dismissal for violent conduct, Alberto Costa levelled in the 88th minute and substitute Nelson Valdez nodded a 90th-minute winner.
Valencia are eighth on 11 points while Bilbao are 16th.
Valdez scored with his head into the lower right of the goal after receiving a pass from Tino Costa.
Other goals in the match came on a right-footed shot by Aritz Aduriz in the 19th minute, a right-footed shot by Roberto Soldado in the 26th minute, a right-footed shot by Aduriz in the 30th minute and a right-footed shot by Tino Costa in the 88th minute.
Los Che pounded Los Leones with shots, attempting 27 shots during the match, with 12 on target. Valencia had all these opportunities while controlling possession for 54.6% of the match. The most effective of Xotos's three substitutions was Valdez, who, after replacing Jonas in the 62nd minute, scored one goal and took five shots.
Athletic Bilbao saw their chance of victory slip away after Ander received a red card in the 67th minute. Los Rojiblancos were forced to play out the rest of the contest with 10 men as a result of the straight red, which was handed down for fighting. Athletic Bilbao attempted four shots, including three on target.
Midfielder Tino Costa was superb both going forward and defending for Valencia. In addition to his five shots, he also blocked two Los Leones scoring attempts and won eight duels.
Athletic Bilbao finished the first half having controlled the ball for 55.7%, with Los Che at 44.3%. Athletic Bilbao led at half-time even though they only attempted six shots compared to Valencia's 10.
Los Murcielagos now sit in eighth in the table with a record of three wins, three losses and two draws. Rojiblancos are in 16th place with a record of two wins, four losses and two draws.
- Aritz Aduriz
- Nelson Valdez
- Tino Costa
- Athletic Bilbao
- Roberto Soldado