Williams is considered one of the most exciting young talents at Bilbao and the 21-year-old showed what he is capable of with two superb goals at a rain-lashed Benito Villamarin.
He opened the scoring in the eighth minute with a powerful drive from the edge of the penalty area before flicking a Raul Garcia centre past Antonio Adan from close range to make it 2-0 moments before halftime.
Williams had a busy night and conceded a penalty for handball that allowed Ruben Castro to pull a goal back in the 67th minute.
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Betis were unable to equalise despite sustained late pressure and Garcia struck a sweet shot low into the corner three minutes from time to make the game safe and secure a first away win of the season.
Bilbao, who host Partizan Belgrade in the Europa League on Thursday, lost four of their first five games of the La Liga campaign but have won three of their last four and have 14 points from 10 matches.
They are a point behind seventh-placed Valencia and two ahead of 13th-placed Betis, last season's second-division champions.
Eibar climbed to sixth when the modest Basque club edged visiting Rayo Vallecano 1-0, while Espanyol managed to conjure a last-gasp equaliser at home to Granada despite having to play the last 20 minutes with 10 men.
Sporting Gijon won 1-0 at home to struggling Malaga thanks to a first-half effort from Croatia teenager Alen Halilovic, who is on loan from champions Barcelona.
Real Madrid and Barca took advantage of Friday's stumble by Atletico Madrid to surge three points clear at the top as they beat Las Palmas 3-1 at home and won 2-0 at Getafe respectively on Saturday.
The Spanish heavyweights each have 24 points from 10 matches, with Real ahead on goal difference.
Celta Vigo climbed above Atletico, who conceded a late equaliser in a 1-1 draw at Deportivo La Coruna, into third on 21 points with a 3-2 comeback win at Real Sociedad on Saturday.
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