Oihan Sancet gave Athletic a second-half lead before Dani Parejo scored in injury-time for Villarreal.
"It was a tough match, we competed well," said Valverde.
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"Both of us were going hard for the win. After seeing how difficult it was for us, given the week we've had, I think the team responded well."
He added: "We have to keep fighting to improve our position. The idea is to take points off those in front."