Well the Dark Ages weren't as dark as people like to claim. In fact many Medievalists cringe at the word "dark ages." While it is true that the Roman Empire (based in Rome) fell, in reality the usurpers of political power in the Western Roman world tried their best to emulate Roman culture and be as "Roman" as they could. Visigoths, Franks, and Ostrogoths all viewed themselves as legitimate successors of Rome, her culture, and her identity. Not to mention the Franks had in fact been Roman allies based in Toxandria around the city of Tournai in northern Gaul.
Also, many monasteries and churches kept classical manuscripts, though many times these monasteries fell prey to vandals and eventually Vikings and other marauders. Sometimes clergy would write on top of an old classical manuscript, though these writings are decipherable. With the Carolingian Renaissance in the 8th-9th century, Frankish emperors (who called themselves Roman Emperors after Charlemagne) commissioned the copying and preservation of many old classics. If you read the "Life of Charlemagne" by Charlemagne courtier, Einhard, you'll see just how the classics had already permeated into literature- Einhard makes references to classical writings left and right, not to mention he practically plagiarized from earlier biographies of Augustus when writing Charlemagne's biography.
Well I simplified a few things for this post but that's the gist of why the "Dark Ages" weren't as dark as some people would like to believe.
EDIT: ^^and yes, especially for Ancient Greek writings, Arabic translations and copies were critical to their preservation. My post was mainly regarding Latin manuscripts.