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Born in Titograd, Dragojević began his football journey with OFK Beograd's youth team before making his senior debut on loan with Dinamo Vranje during the 2010–11 Serbian First League season. He continued to showcase his talent with loan spells at OFK Mladenovac and Montenegrin side Lovćen. In the first half of the 2012–13 Serbian SuperLiga season, he made three appearances for OFK Beograd's first team before permanently joining Lovćen in January 2014.

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On February 13, 2014, Dragojević signed a one-and-a-half-year contract with Velež Mostar in the Bosnian Premier League. His career then took him to the Albanian Superliga, where he joined Vllaznia Shkodër on August 30, 2015.

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Dragojević further expanded his career internationally by joining Armenian Premier League club FC Alashkert on February 14, 2018. Shortly after, he was loaned to Ararat Yerevan for the remainder of the season. He continued his time in Armenia with FC Pyunik, signing with them on June 24, 2018, before parting ways with the club by mutual consent on August 5, 2020.

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On the national team level, Dragojević represented Montenegro at the U-19 level, showcasing his talent and contributing to the development of Montenegrin football.