Dortmund scored in the 36th minute through Niclas Füllkrug, who went on the edge of offside to beat Gianluigi Donnarumma.
The second leg will be played next Tuesday at the Parc des Princes.
In the second semi-final in the first half, Real Madrid and Bayern Munich met to a draw (2-2) on Tuesday evening. The two teams meet next Wednesday in the second semi-final.
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