The defeat left Sevilla third in the standings on 33 points, seven behind leaders Barcelona who could stretch their advantage over the Andalusians to 10 if they beat Eibar later on Sunday.
Athletic meanwhile climbed up to 16th on 22 points, moving three away from the relegation zone.
Williams gave them the lead midway through the first half, cutting in from the left wing to find the far top corner with a scorching strike -- his first goal at home since netting against Eibar in December 2016.
Sevilla had beaten Athletic 3-1 at San Mames three days previously in a Copa del Rey last-16 game but for most of the game were outmuscled by Gaizka Garitano's side, who are fighting against a potential first-ever relegation from the top flight.
Dutch midfielder Quincy Promes came closest to scoring for the visitors with a long strange strike which brushed past the far post, while Argentine playmaker Ever Banega saw a shot squirm just wide from the edge of the area.
But Williams, who also scored the winner at Celta Vigo last week, sealed the victory with another impressive effort in the 82nd minute, knocking the ball beyond a defender and then dribbling around the goalkeeper before tapping in.
(Reporting by Richard Martin; Editing by Ian Chadband)
