For the first time in 10 years, neither Barcelona nor Real Madrid will participate in the cup semis.
Thursday's match in Bilbao was a largely humdrum affair before Williams' winner sparked hysteria.
Barcelona dominated with 68% possession but were strangely poor, with second-half substitute Antoine Griezmann and Lionel Messi both thwarted by Athletic shot-stopper Unai Simon during the match's latter stages.
Athletic's Unai Nunez also made a pair of vital interventions, first in blocking an Ansu Fati effort before denying Messi later in the second stanza.
There were few other clear-cut chances, with Williams' correctly ruled offside after slotting home from close range.
Barcelona's Frenkie de Jong also had a decent opportunity when a delightful ball from Messi gifted the Dutch midfielder the opportunity to round Simon. Instead, De Jong went to ground, sensing contact from the 'keeper, and was shown a yellow card for diving.
Quique Setien made just a single change to the XI that beat Levante at the weekend, with Sergi Roberto coming in for Griezmann.
The semifinal draw takes place Friday, Feb. 7, and offers the tantalizing prospect of an all-Basque final.