Williams' first goal since February snatched a 1-0 win for Athletic and leaves Sevilla six points adrift of leaders Atletico Madrid, with four games left to play.
Sevilla's first of those will be against Real Madrid on Sunday at Valdebebas, the second part of a potentially pivotal weekend in the title race, after Barcelona go up against Atletico on Saturday.
Atletico are two points clear of Real Madrid and Barca after they beat Elche on Saturday, before Real Madrid saw off Osasuna and Barcelona edged past Valencia on Sunday night.
Sevilla had been the form team at the top, having won five games in a row, but their chances of overhauling the top three now look slim.
They will surely have to beat Real Madrid next weekend to reignite their hopes, before finishing the season at home to Valencia, away at Villarreal and at home to Alaves.
Williams held his nerve at the crucial moment, sent clear after Oihan Sancet had torn away on the break before slipping left to his team-mate.
With Sevilla's Fernando pulled across, Williams had time to pick his spot, opening up his right foot and lifting the ball into the right-hand corner.
Victory lifts Athletic Bilbao to ninth place, six points behind Real Betis in seventh.