The London club largely controlled the match but rarely looked like scoring and conceded a goal in each half thanks to two mistakes in the Estadio Alfredo di Stefano.
Arsenal began with Steph Catley in defence and Kyra Cooney-Cross in midfield. Caitlin Foord joined them when she was introduced at half-time.
In teeming rain on a dreadful, waterlogged pitch Arsenal struggled to assert themselves and went behind after 22 minutes. After a slack Cooney-Cross pass was cut out England captain Leah Williamson allowed Signe Bruun's forward ball to Linda Caicedo to run under her foot and the Colombian punished her with a cool finish.
Either side of the goal Stina Blackstenius and Beth Mead went close but they were rare forays forward and both were replaced at half-time.
Foord was among the arrivals and was soon running at the home defence, having a shot after 53 minutes. Her directness, and that of fellow sub Chloe Kelly, made a difference and both tested the Real defence.
But as it seemed Arsenal were sure to level possession was again conceded cheaply in midfield. Athenea del Castillo was then allowed to shoot from the edge of the box. Manuela Zinsberger should have saved, but allowed the ball through her palms into the net.
With Arsenal eight points behind leaders Chelsea in the Women's Super League and out of both domestic cups, Europe represents their last realistic chance of winning a trophy this season.
In the night's later match Lyon, managed by Joe Montemurro and featuring Ellie Carpenter, are at Bayern Munich. Carpenter is the only Australian women to win Europe's premier club prize.
On Wednesday Chelsea, who remain without Sam Kerr, are at Mary Fowler's Manchester City. Fowler has added responsibility as City will be without injured forwards Khadija Shaw and Aoba Fujino.
The same night Wolfsburg host defending champions Barcelona.
The second leg ties are next week with Arsenal hosting Real at the Emirates on Wednesday.
