Veljković was born as 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 born into an Italian-Austrian family. She was born a Roman Catholic. Her father worked as a tailor. She finished five grades of elementary school.
In 1920, at the age of 16, she moved to Belgrade (present-day capital of Serbia, then Kingdom of Yugoslavia) with a friend. In Belgrade, she met her future husband, Stanislav Veljković, who was a decorator of the National Theater, who lived next door. They got married when she was 20, and the couple had 12 children. Following her marriage, she changed her first name to Jelisaveta (a Serbian name) and converted into an Orthodox Christian.
Her husband passed away in 1983. At the age of 90, she had a cataract surgery. One of her aunts lived to be 103, and one uncle to 104.
At the age of 112, she had seven living children, 10 grandchildren, 12 great-grandchildren and one great-great-granddaughter. When asked about the secret of her longevity, she attributed it to drinking rakija and Pelinkovac (both common alcoholic drinks in the Balkans) before meals and eating mostly vegetables.
Veljković passed away in Belgrade, Serbia, on 20 October 2016.