Malaga drew 1-1 at AC Milan to progress to the knockout stages of Europe's elite club competition but goals on the counter attack from Mexico striker Carlos Vela and Xavier Prieto condemned Manuel Pellegrini's men to a second consecutive home defeat.
Bottom club Osasuna earned their first points on the road this season with a 3-0 victory at fellow strugglers Espanyol on Saturday.
A lucky deflection and a composed finish from Prieto restored Sociedad's lead after 59 minutes and they held on for a first away win since January.
In Barcelona, Osasuna's Alvaro Cejudo netted a scrappy opener and after the hosts had twice hit the woodwork, Kike Sola dribbled round the keeper to make it 2-0 in the 63rd minute.
"If I was hanging by a thread before, I still am now," Mendilibar told a news conference. "This isn't about one day, it is a marathon and there are many kilometres left to run."
Rayo made the most of a harsh second yellow card for Celta's Gustavo Cabral just before half-time, with a stirring second half display in Vallecas.
Leo Baptistao headed Rayo's first and Tito nodded the equaliser in the 72nd minute before a disputed penalty allowed Piti to claim the winner from the spot.
Deportivo's week went from bad to worse after announcing on Thursday they were on the brink of entering administration, as they squandered an early two-goal lead in Zaragoza.
