Mikel Oyarzabal scored the first of his two spot-kicks on the half hour mark after former Sociedad defender Inigo Martinez was judged by the video assistant referee (VAR) to have kicked an opponent while contesting a high ball.
Athletic responded immediately with an equaliser from Iker Muniain, who began the move with a fine piece of skill on the halfway line and finished it by belting the loose ball home from close range after it had cannoned off two Sociedad defenders.
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The visitors re-took the lead two minutes into the second half when Luca Sangalli slid in to meet a cross from Igor Zubeldia, who had won the ball back in midfield and danced his way down the right wing.
Sociedad killed the game off when Oyarzabal struck again from the spot in the 74th minute to leave Bilbao 16th in the table with seven points, four adrift of Sociedad.
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Suarez nets twice to rescue point for Atletico against Real Sociedad