The LaLiga side played almost 80 minutes with 10 men, as the referee sent off Sime Vrsaljko for two bookable offences.
Arsenal failed to add a second leg, before Antoine Griezmann pounced on a hopeful long ball and scored a crucial away goal for his team.
The second leg comes up next Thursday at the Wanda Metropolitano in Spain.
In the other semi-final, Marseille claimed a comfortable 2-0 win over Salzburg at home.

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