Real looked to have blown their chances when they threw away an early two-goal advantage, but a scrappy goal by Diego Llorente restored their lead in the 83rd minute before Asier Illarramendi made sure of the points in the 86th.
Juanmi Jimenez put the visitors ahead in the third minute with a volley following a quickly-taken free kick and less than 60 seconds later they scored again as Illarramendi appeared on the edge of the area to arrow the ball low into the far bottom corner.
Deportivo responded with a well-struck volley from former Atletico Madrid forward Adrian Lopez midway through the first half and drew level five minutes into the second when Florin Andone levelled from close range.
Former Real Madrid youth player Llorente stuck out a leg to take advantage of some shoddy defending from Deportivo at a corner to put Real Sociedad ahead again.
Illarramendi then scored his second, the first time in his career that he has scored twice in a game, to ensure the Basques maintained their perfect start.
Next up for Eusebio Sacristan's side is a home game next Sunday against champions Real Madrid, who have taken five points so far and were held to a shock 1-1 draw at home to promoted Levante on Saturday.
(Reporting by Richard Martin; Editing by Toby Davis)
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