Babović was born in Ivangrad, SFR Yugoslavia (present-day Berane, Montenegro) but grew up in Belgrade where he came through the youth ranks at Partizan and made his senior debut at the age of 17, coming on as a substitute in a 3–0 home win over Sutjeska Nikšić on 17 March 2004.

[5][6] After parting ways with Partizan, Babović joined OFK Beograd as a free agent, penning a four-year contract.

[10] During his first six months at the Stade de la Beaujoire, Babović made 15 league appearances, scoring no goals, as the club won promotion to the Ligue 1.

On 1 September 2009, after featuring in Nantes' first three league matches of the campaign, Babović was sent on loan to Feyenoord until the end of the 2009–10 season,[11] with a buyout clause.

[13] After serving his suspension,[14] Babović returned to the lineup for the match against Groningen and scored his first Eredivisie goal, an equalizer in a 3–1 victory.

In European competitions, Babović recorded four appearances in the group stage of the UEFA Champions League, as the club finished bottom of the table.

[21] On 21 February 2015, Babović made his third debut for Partizan, playing the full 90 minutes in a home league fixture against his former club Voždovac; the game ended in a 3–3 draw.

[23] On 14 July 2015, Babović netted his first goal in UEFA competitions, scoring from a free kick to give his team a 1–0 win over Dila Gori.

He was subsequently Serbia's top scorer during the 2007 Championship qualifying stage; two of his three goals came in the stunning play-off comeback against Sweden.

After appearing in an unofficial friendly against the Basque Country in December 2006,[29] Babović made his official debut for the senior national team under Javier Clemente in a rescheduled UEFA Euro 2008 qualifier against Kazakhstan on 24 November 2007.

[30] In November 2017, Babović was appointed chief operations officer (COO) of Serbian holding company Victoria Group.