Arsenal, who avenged a 2-1 loss to Dortmund in London last month, bided their time for more than an hour in pouring rain, soaking up Dortmund attacks before striking with their first effort on goal.
The visitors scored against the run of play with Wales international Ramsey sneaking into the box to head home in the 62nd minute. The result lifts Arsenal to nine points from four games with Dortmund on six.
Dortmund, who until Wednesday had never lost at home to English opposition in seven contests, wasted half a dozen clear chances but failed to make their dominance count.
(Reporting by Karolos Grohmann; Editing by Clare Fallon)
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