Celta's talismanic forward Iago Aspas scored the only goal of the game from the penalty spot on the stroke of halftime after being felled in the area by Diego Llorente, who also gave away a spot-kick in Sunday's 2-1 defeat to Real Madrid.
The penalty was Celta's only shot on target in the 90 minutes and they spent the second half sitting back and defending their lead, with Colombian defender Jeison Murillo producing an outstanding display to help shut out the hosts.
The victory followed Sunday's 6-0 thrashing of Alaves and took Celta, who are 16th, seven points clear of the relegation zone. Sociedad's top-four hopes meanwhile are fading fast as they sit in seventh, six behind fourth-placed Sevilla.
Sociedad badly needed a win to kick-start their bid to return to the Champions League places after a wobbly run since the season returned from a three-month pause due to the COVID-19 pandemic.
They were reeling from their defeat by Madrid in which key refereeing decisions went against them and they again found themselves penalised when Aspas went down after having his shirt tugged by Llorente.
The Basque side dominated the second half but struggled to create much, stifled by Celta's line of five defenders, and slumped to a third straight defeat, picking up only one point in four matches since the season resumed.