Born in Belgrade, Aničić started youth career in FK Crvena Zvezda in 1979.

After that he coached FK Sloga Petrovac na Mlavi,[1] FK BSK Borča[2] and finally in January 2017 he was appointed head coach of FK Mačva Šabac,[3][4] leading the team to promotion from Serbian First League to Serbian SuperLiga after 65 years playing in lower divisions[5][6] On 15 October 2018, Aničić was appointed head coach of FK Voždovac.

[7][8][9][10][11] In September 2019, Aničić returned to FK Mačva Šabac as manager and stayed till December 2020.

[12] In January 2021, Aničić was appointed head coach of FK Železničar Pančevo[13][14][15] In September 2022, Aničić was appointed head coach of FK Rudar Pljevlja[16][17][18] In June 2023, Aničić was appointed head coach of FK Novi Pazar[19][20][21] In June 2024, Aničić was appointed head coach of GFK Sloven Ruma[22][23] In August 2024.

Dragan Anicic left GFK Sloven Ruma and accepted an offer from Aleksandar Veselinovic to form part of the coaching staff of the Albanian Kategoria Superiore team FK Partizani Tirana.