Antoine Griezmann put the hosts ahead in the closing stages of the first half and although Ernesto Valverde's side had significantly more possession, they were unable to carve together enough chances to equalise, and in injury time Ruben Pardo provided the finishing touch to a counterattack to seal the points.
Athletic went into the game four points clear of their rivals in fourth and the game had been billed as a crunch tie in the battle for the final Champions League spot.
The visitors made the brighter start, with Ander Herrera and Aymer Laporte both shooting off target early on. Carlos Vela had La Real's first notable effort and then Imanol Agirretxe came close to flicking a cross from the Mexican into the goal.
At the other end, Carlos Gurpegui headed just wide of the near post from close range following a corner, and later Aritz Aduriz hit the side netting.
Valverde's men looked the likelier team to score but with little more than two minutes to play of the first half, it was La Real who took the lead.
Pardo sent in an out-swinging corner towards Griezmann on the left hand side of the area, and the winger's shot was kicked out by Athletic stopper Gorka Iraizoz straight back to the Frenchman, who struck first time and this time the ball bounced in at the near post.
The goalkeeper's view had been blocked by the outstretched leg of Vela, but the former Arsenal striker did not get a touch, and Iraizoz will feel he should have done much better.
La Real started the second period on the front foot and could have doubled their lead through one of two own goals. First Mikel Rico flicked a Griezmann corner just wide of the near post, then Mikel Balenziaga headed a corner from Pardo off the post.
Athletic soon improved but were imprecise when they got to their hosts' area, and had appeals for a handball by Markel Bergara in the area turned down.
Midway through the second half Griezmann was put through on a rare break for La Real, but this time Iraizoz got the better of his opponent. For all their possession, Athletic created few clear cut chances.
Herrera should have done more when presented with a free shot on the edge of the area, which he struck into the arms of Claudio Bravo, while Aduriz came inches away from sending a cross from the left hand side inside the near post.
Athletic became more and more desperate as time ran out and left their left hand side exposed, allowing Pardo to stroll into the area, receive a pass from Haris Seferovic and calmly slot into the opposite corner.