Milenković began studying the violin with his father when he was three. Three years later, he made his debut as a soloist with an orchestra, and started touring extensively in 1984. In 1987, he performed for U.S. President Ronald Reagan at a Christmas concert in Washington, DC. The following year, he played for Mikhail Gorbachev in Belgrade. In 1991, he played for Pope John Paul II, and in 1993, Milenković gave his 1000th concert in Monterrey, Mexico.
After graduating from the University of Belgrade in 1994 (its youngest ever graduate), he was invited to study with Dorothy Delay at Juilliard. He continued to tour with major orchestras and was named a faculty member of the Perlman Music Program in 2000. In 2003, he was named a faculty member of Juilliard's College and Pre-College Divisions, where he also serves as Itzhak Perlman's teaching assistant. One of his notable students there was Ryu Goto, now an internationally recognized violinist.