Veljković was born Vicenca (Vicencija) Kufner in Ravna Gora, Austria-Hungary (now in Primorje-Gorski Kotar, Croatia) on 18 July 1904. She was the second of seven children in an Italian-Austrian family and was raised Roman Catholic. Her father worked as a tailor. She completed five grades of elementary school.
In 1920, at the age of 16, she moved to Belgrade—then part of the Kingdom of Yugoslavia—with a friend. There, she met her future husband, Stanislav Veljković, a decorator at the National Theater who lived next door. They married when she was 20 and went on to have 12 children. After her marriage, she adopted the Serbian name Jelisaveta and converted to Orthodox Christianity.
Her husband died in 1983. At the age of 90, she underwent cataract surgery. Longevity seemed to run in her family—one of her aunts lived to 103, and an uncle to 104.
By the age of 112, she had seven living children, 10 grandchildren, 12 great-grandchildren, and one great-great-granddaughter. When asked about the secret to her long life, she credited drinking rakija and Pelinkovac (both popular alcoholic drinks in the Balkans) before meals, and eating mostly vegetables.