Published: 2020-02-07 05:03:09 BdST
Spanish striker Williams glanced a cross into the bottom corner to spark jubilant scenes at a rocking San Mames stadium and recalled Aritz Aduriz's added-time goal in Athletic's 1-0 win at home to Barca on the first day of the Liga season.
They join Real Sociedad, Granada and second-tier Mirandes in Friday's semi-final draw, which will not include Real Madrid, Barca and Atletico Madrid for the first time in 17 years.
Coach Quique Setien said he was hurt by the elimination but was pleased with his side's performance.
"Today everything went well apart from the result," he said. "We took a step forward in our play and I'm pleased with many things we did, I'm not only looking at the result."
Barca captain Lionel Messi spurned a glorious late opportunity for the visitors when he could only shoot at the legs of Athletic keeper Unai Simon while his fellow striker Antoine Griezmann also came close in the second half.
The game looked to be heading into extra time when Williams missed a gilt-edged chance inside the area but he made amends by getting the slightest of touches on a cross from Ibai Gomez in the third minute of added time.
"I'm lost for words, you should never lose faith," Williams told reporters. "The fans deserved this moment after the way they supported us and I now hope we can go on and fulfil the dream that we all have by going on to win it."
The victory must have also felt like redemption for Athletic, who were knocked out of the Cup by Barca in 2017 and 2016 and lost to them in the final in 2009, 2012 and 2015.
The Catalans have appeared in the last six finals but they join Real Madrid in being knocked out after Zinedine Zidane's side were beaten 4-3 at home to Real Sociedad earlier on Thursday.