Benat Etxebarria then curled a free kick over the crossbar in the 18th minute and Valencia made the most of their reprieves by taking the lead on 23 minutes.
Despite their spurned first-half opportunities, Jose Angel Ziganda's charges regrouped and drew level on 49 minutes.
Oscar de Marcos showed good technique to produce a first-time effort from just outside the box that beat Neto and nestled inside the right post.
From then on the better chances fell to the Basques, but Aduriz and Raul Garcia failed to hit the target from good positions as the final whistle blew on a draw.
The result means that Valencia remain fourth in the La Liga table, a point and a place behind Real Madrid, while Athletic remain 12th in the standings.
ATHLETIC BILBAO (4-2-3-1): Arrizabalaga; de Marcos, Nunez, Inigo Martinez, Saborit; Benat, San Jose; Susaeta (Williams 63'), García (Lekue 77'), Cordoba; Aduriz
VALENCIA (4-4-2): Neto; Vezo, Gabriel, Murillo, Gaya; Torres (Guedes 75'), Parejo, Kondogbia, Coquelin; Vietto (Rodrigo 71'), Zaza (Soler 83')