Depor took a 27th-minute through Luis Pizzi but then found themselves on the back foot after conceding an equalizer to Carlos Vela on the stroke of half-time and then losing defender Evaldo to a red card just after the hour-mark.
However, Sociedad could not make the most of their numerical advantage in the closing stages as Depor made it four points from two games under new coach Domingos Paciencia.
Depor downed high-flying Malaga in Domingos' first match to climb off the foot of the standings and they put themselves on course for another impressive scalp when Pizzi breached a Sociedad backline with the best defensive home record in La Liga this season to fire the visitors ahead.
However, mid-table Sociedad, who had only conceded five times at the Anoeta previously, hit back and almost leveled within 10 minutes as Imanol Agirretxe headed against the post.
That was a big let-off for Depor but they were not so fortunate in the 45th minute as Vela tucked the ball home from close range to make it 1-1.
The start of the second half saw a spate of cards, with referee Carlos Del Cerro Grande - having not cautioned anybody in the first half - going to his pocket eight times in a 20-minute spell.
Seven of those were only cautions, but two were for Evaldo and that was enough to earn the Brazilian an early bath and leave Depor facing a battle to avoid defeat.
The Galician side managed to do just that, though, with Sociedad forwards Antoine Griezmann and Vela unable to make the most of the chances that fell their way.