Man-of-the-match Muniain scored the winner with a penalty after the video assistant referee spotted a handball from Jordi Alba inside the area.
Barcelona's disappointing campaign took another turn for the worse.
Eliminated at the Champions League group stage, they lost to Bilbao in a repeat of last year's final which they won 4-0 and two of their most promising players, Ansu Fati and Pedri, were substituted due to injury.
Asked if their elimination from the competition is a failure, Xavi said: "I don't like the word failure because we tried. But if we have failed here then let it be something to learn from ... we learn from our mistakes and try again."
"We have two battles left: Europa League and LaLiga. We fought until the end but in many moments Athletic bettered us with intensity, they were better than us, have to congratulate them. San Mames [is a tough place to play], especially with this Athletic team and this coach. They're a tough opponent."