Eskenazi was born Sarah Skinazi to an impoverished Sephardic Jewish family in Istanbul. Throughout her career she hid her real date of birth, and claimed to have been born in 1910. In fact, she was at least a decade older, and was likely born sometime between 1895 and 1897. Her father, Avram Skinazi, was a rag dealer. In addition to Roza, he and his wife Flora had two sons, Nisim, the eldest, and Sami.

Shortly after the turn of the century, Skinazi family relocated to Thessaloniki, then still under Ottoman rule. The city was undergoing rapid economic expansion at the time, with its population growing by 70 percent between 1870 and 1917. Avram Skinazi found work in a cotton processing mill and took various odd jobs to improve his family’s financial standing. At the time, he entrusted young Sarah to a neighboring girl, who tutored several local children in basic reading and writing. These sessions were the extent of her formal education.

For some time, Sarah, her brother, and her mother lived in nearby Komotini, a city that at that time, still had a sizable Turkish-speaking population. Roza's mother found employment there as the live-in maid for a wealthy family, and Roza assisted her with the housework. One day, Sarah was overheard singing by the Turkish owners of a local tavern. They were enthralled by her voice, and immediately came to the door to express their wish to hire the girl to perform in their club. Sarah's mother was incensed at the suggestion that her daughter—or any other member of her family—would become an artiste. Years later, in an interview, Roza admitted that her time in Komotini was a turning point in her life. It was there, she said, that she decided to become a singer and dancer.