The European champions seemed set for a comfortable night on Sunday when early goals from Sergio Ramos and Gareth Bale left the home side shell-shocked.
But Sociedad regrouped outstandingly, with goals from Inigo Martinez and Carlos Vela complementing Zurutuza's brace to seal a famous triumph.
Questions have surrounded Carlo Ancelotti's Real side in 2014-15 so far after an uninspiring 2-0 win over Cordoba last Monday followed their Supercopa de Espana defeat to Atletico Madrid.
But having found themselves 2-0 ahead after just 11 minutes, Ancelotti's men let Sociedad back into the game and were made to pay the heaviest of prices.
Real flew out of the blocks and took an early lead when Ramos rose to meet Toni Kroos' corner with a glancing header at the front post.
And Real did double their advantage in the 11th minute as Bale brilliantly nutmegged Gorka Elustondo and slotted a cool finish into the corner.
Not content with being 2-0 up, Real continued to press forward - only to see Marcelo twice spurn good chances before Ramos headed just over from a James Rodriguez set-piece.
The defender was pressed into action in his own box soon after, bravely flinging himself in front of Esteban Granero's powerful, goal-bound effort.
And the scores were level six minutes later as Zurutuza arrived late into the box to powerfully head home Alberto de la Bella's centre.
Real's defending was not much better, illustrated at the start of the second half when an unmarked Imanol Agirretxe saw a header turned wide by the toe of Iker Casillas.
The verve with which Real attacked early on seemed a distant memory as they struggled for cutting edge.
Yet it was the issues at the other end that were most troubling, their inability to clear a set-piece proving costly when Vela followed a fine piece of chest control with a volley that arrowed into the top corner.
Sami Khedira, Rodriguez and Karim Benzema all missed golden opportunities thereafter as Real failed to find a route back into the game.
