Junuzović moved with his parents to Austria when he was five years old. Before his 12th birthday, they went on to Graz, where he soon played for the youth teams of Grazer AK.
In 2010, Junuzović was awarded two awards for his performances with Austria Wien and the national team: the managers of the Austrian Bundesliga voted him Austrian Footballer of the Year[7] and the country's supporters voted him "Player of the Year"[8] in the annual award awarded by the Kronen Zeitung.
In January 2012, Junuzović joined Bundesliga club Werder Bremen, signing a contract which lasted until 30 June 2015. In February 2015, it was announced that his contract had been extended until June 2018.[9]
Having completed the 2014–15 season with six goals and 15 assists, third in the Bundesliga behind Kevin De Bruyne and Thomas Müller, Junuzović was voted as "Player of the Season" by Werder's fans.[10]
On 18 March 2017, he scored his second goal of the 2016–17 season in a 3–0 win against Leipzig, a match that saw all three of Werder Bremen's Austrian players score in the club's highest win of the season.[11][12]
In April 2018, Junuzović announced he would leave Werder Bremen after six and half years at the club.[13][14] He went to FC Red Bull Salzburg in the Austrian Bundesliga. In 2022, he ended his career and became assistant coach of FC Liefering.[15]
In October 2017, after the conclusion of the qualifiers for the 2018 World Cup during which Austria failed to reach qualification, Junuzović announced his retirement from the national team.[18]