Sexton graduated with an MBBS in 1892, making her the third woman graduate from the University of Melbourne's medical school. Since most hospitals were reluctant to hire female doctors, Sexton joined a group of women, led by Constance Stone, who co-founded the Queen Victoria Hospital for Women and Children in 1896. When the hospital opened in 1899, Sexton was appointed the head of surgery, a position she held until 1908.