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The visitors were handed the chance to draw level almost immediately when Fernando Navarro brought down Iago Aspas in the area and a penalty was awarded. However, Nolito's spot kick towards the bottom corner was palmed to safety by Lux.
Jonathan Rodriguez almost doubled the home side's advantage as he raided to the edge of the box and cut back on to his right foot, but his strike sailed inches over the crossbar.
In the second half, Alvarez pulled off a fine stop to keep the visitors in the game by denying Rodriguez as he saw his powerful strike blocked by the Celta keeper.
Juanfran came within inches of securing the win for Deportivo in the 76th minute when he latched on to a pinpoint pass inside the penalty area and powered an effort towards the net, only for his strike to cannon of the post.
Aspas almost unlocked the home side's defence in the dying stages of the match as he weaved his way past two defenders, but Lux made a good save with his body to block his shot.
Celta piled men forward in the final seconds of the match in a desperate attempt to salvage a point, but they were left open at the back and the home side secured the three points when Jonny's clearance back to Alvarez sailed past the keeper into the net.
The result moved Deportivo into eighth place in La Liga, while Celta remain in fourth.