Nico got an assist in each half and Iñaki scored twice before the team was reduced to nine men and coach Ernesto Valverde red-carded.
Friday's win resulted in Athletic closing the gap to three points behind fourth-placed Atletico Madrid.
The top four in La Liga go into the Champions League. Atletico have five games to play, including at Mallorca on Saturday. Athletic have four games remaining.
As has happened so often this season, the Williams brothers stood out for the Basque club.
With 26 minutes gone, Niko played a short pass to Iñaki and from 20 metres out he curled the ball in.
Six minutes into the second period, Nico played a defence-splitting pass to his brother who swept the ball home with aplomb.
Athletic were then punished for indiscipline. Yeray Álvarez was sent off for handball after 58 minutes and fellow defender Aitor Paredes joined him 11 minutes from time.
Getafe threw men forward and was awarded a penalty with two minutes remaining but Mason Greenwood, one of the home side's stand-out performers, saw his spot kick saved by the outstanding Unai Simón.
Athletic coach Valverde was shown a red card for protesting against the spot-kick decision.
Getafe are in 10th place.
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