Sergio Ramos headed in a corner after five minutes followed by a delightful effort from Gareth Bale who slipped the ball between Gorka Elustondo’s legs inside the area.
It appeared comfortable for Real as they continued to press forward but the home side battled back as the first half progressed.
Inigo Martinez swept home a cross at the far post after 35 minutes and Zurutuza was left unmarked to head the second.
After the break Zurutuza again and substitute Carlos Vela took further advantage of defensive lapses to seal a convincing win over the European champions.
Earlier, substitute Sandro Ramirez slotted home eight minutes from time to continue Luis Enrique’s winning start as Barcelona coach with a 1-0 victory over Villarreal.
Barca were frustrated by opponents who defended well but the introduction of fit-again forward Neymar for the final half hour gave them an extra attacking edge.
It was Lionel Messi, though, who opened up the Villarreal rearguard with a cross along the goalline that was turned in by Ramirez to lift Barca to six points from their opening two games.
"I am happy on a personal level but above all for the work of the team," Ramirez told reporters. "Thanks to Leo I could score the winning goal. Messi and Neymar are fantastic players although this is the work of all of us and we are all happy."
Enrique played the promising Munir El Haddadi, who scored last weekend, ahead of Neymar in Barcelona's starting line-up.
With Andres Iniesta sidelined by a knee injury, Rafinha played in midfield rather than veteran Xavi who is set to have a fringe role in the team this season.
Barca had the majority of the ball but struggled to find space going forward against a well-organised Villarreal side and it was only following Neymar’s entry that the chances began to build up.
Messi hit the post with a free kick from the right of the area in the first half but Barca were also lucky when Jeremy Mathieu turned a cross onto an upright.
Tomas Pina and Messi again hit the woodwork for either side midway through the second half before Neymar found Messi, who looked up and picked out Sandro to score and give the Catalan side the three points.
(Editing by Ed Osmond/Alan Baldwin)
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