The Gunners completed a smash and grab job in the Champions League to edge last season's runners-up and remain top of the Group F table.
Just after the hour-mark, Mesut Ozil played a looping ball into the area which was nodded into the path of Ramsey by Olivier Giroud for the Welshman to guide past Roman Weidenfeller for his fifth tally in Europe this season and his 11th goal in all competitions.
Dortmund had gone into the game having won their past seven matches at home but the Gunners overcame them to make it four wins on the trot away from home in Europe, which extended their unbeaten streak on the road to 18 matches in all competitions.
Arsenal now share top spot with Napoli on nine points, after the Italians beat bottom-placed Marseille 3-2 in Naples, whilst Dortmund sit third on six points with two matches remaining in the group stage.
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