Begić began his career playing with Sloboda Dita Tuzla in Bosnia and Herzegovina. He made his professional debut with Sloboda in the Bosnia and Herzegovina League during the 2001–02 season. He then arrived to Slovenia in 2002, and stayed in KK Koper for 2 months. In the 2002–03 season, he played for Union Olimpija's junior team. Then he moved to Triglav Kranj in the Slovenian League. Begić averaged 7 points, 5 rebounds, and 1.7 blocks, in 16.9 minutes in his first season in the Slovenian League in 2003–04. In 2004, Begić signed with the Italian League club Virtus Bologna. In his first season with the club he was loaned to Huy Basket in Belgium's First Division. He averaged 6.3 points and 4.2 rebounds per game. In the 2005-06 season, he was released by Virtus Bologna. In the 2006–07 season, Begić moved back to Slovenia. He signed with Geoplin Slovan. He averaged 9.8 points, 6.0 rebounds, and 1.9 blocks in 22.2 minutes per game in the Adriatic League, and 9.3 points, 5.7 rebounds, and 1.5 blocks per game in the Slovenian League.
On July 30, 2007, Begić signed a 2-year contract with Union Olimpija. He made his Euroleague debut on October 24, 2007, versus the Italian League clubMontepaschi Siena. Begić averaged 3.6 points and 2.8 rebounds per game in his first season of the Euroleague play. In his second season in Euroleague, he averaged career-high 11.1 points and 6.1 rebounds per game. In his second season in the Adriatic League, he averaged 3.9 points and 3.5 rebounds per game. He played in the 2008 Adriatic League Final Four in Ljubljana, where his team, Union Olimpija, lost in the semifinals against Partizan Belgrade. In his third season in the Adriatic League, he finished with averages of 8.9 points and 4.5 rebounds per game. He won the Slovenian National Championship and Slovenian Cup with Union Olimpija in 2008 and 2009. On November 24, 2008, he renewed his contact through the end of the 2010–11 season.
On January 18, 2011, Begić reached an agreement with the Spanish League club Real Madrid to play for the club until the end of the 2012–13 season. The 2012–13 season was successful for Begić in Real Madrid, reaching the Euroleague final and winning the Spanish League.
2013–present
On July 20, 2013, Begić signed a two year-contract with the back-to-back defending Euroleague champions Olympiacos of the Greek League. He left Olympiacos after one season. On September 20, 2014, Begić signed a one-year deal with his former team Union Olimpija. On November 19, 2014, he left Olimpija and signed a one-month deal with Laboral Kutxa Baskonia. On December 24, 2014, he extended his contract with Laboral Kutxa Baskonia for the rest of the season. On October 14, 2015, Begić signed with the New Orleans Pelicans, only to be waived by the team two days later. On December 7, 2015, Begić signed with Spanish club Dominion Bilbao Basket for the rest of the season. On July 27, 2016, Begić signed with Croatian club Cedevita Zagreb for the 2016–17 season. On August 13, 2017, Begić signed with Petrochimi Bandar Imam of the Iranian Basketball Super League. On January 17, 2018, he returned to Olimpija.
Begić was born in Bijeljina, SR Bosnia and Herzegovina, Yugoslavia. He lived in his native Bosnia and Herzegovina until 2002, when he moved to Slovenia and subsequently took Slovenian citizenship.
