The Galician club, Spanish title winners in 2000 and Champions League semi-finalists as recently as 2004, have now lost four of their eight matches and drawn the rest and are mired in the relegation places on four points.
An unmarked Antoine Griezmann nodded in Sociedad's second with 20 minutes to go, sparking an unusual celebration from the teenage striker when he jumped behind the wheel of a car on display on the running track around the pitch.
Deportivo have sunk a long way since playing themselves to the brink of a place in the final of Europe's elite club competition six years ago and are suffering the financial problems besetting many of Spain's top clubs.
After finishing 10th last term, they were forced to sell one of their best players, defender Felipe Luis, to Atletico Madrid and had debts of 122 million euros at the end of the 2008-09 season, according to a study by University of Barcelona accounting professor Jose Maria Gay published in May.
Depor's latest defeat leaves coach Miguel Angel Lotina under serious pressure ahead of Thursday's Copa del Rey first leg match at La Liga rivals Osasuna and Sunday's home league match against fifth-placed Espanyol.