According to NSNoob comments on Did Hungarian Army ever use Crescent banners?
Crescent was used by Muslims since the Crusades.
Wikipedia seems to agree. For example: the article on Star and crescent says
The use of the crescent symbol (without the star) on Muslim flags is first seen by Europe during the Crusades.
Wikipedia further listed 14th century examples of flags with crescents: flags of Gabes, Tlemcen (Tilimsi), Damas and Lucania, Cairo, Mahdia, Tunis and Bud; blue ensign of Nubia/Dongola; yellow ensign of the Mamluks. But the Crusades in the Middle East was happening earlier, in 11th-13th century. Briefly looking at the major factions, seems like the Seljuk empire, the Abbasid and Fatimid caliphates and the Ayyubid sultanate don't have crescent in their banners.
Which Muslim factions included the crescent as flags/banners during the initial Crusades of the 11th-13th century? Those that included crescent-and-star are also acceptable.