Dragović started his career by playing in the youth teams of Austrian giantsAustria Wien and started playing for theirB-squad in 2007. In summer 2009 he advanced to their first team, for whom he played 66 championship games. There were also 15 games in the Europa League and 10 games in the Austrian Cup, which he won with Austria in 2009.
On 1 February 2011,Swiss Super League clubFC Basel announced signing Dragović on a four-and-a-half-year contract.[3] He joined Basel's first team during the winter break of their2011–12 season under head coachThorsten Fink. Dragović played hisdomestic league debut for his new club in the home game in theSt. Jakob-Park on 12 February 2011 as Basel won 3–0 againstFC St. Gallen. He was in the starting eleven of every match until the end of the2010–11 Swiss Super League season and Dragović won the Swiss Championship title with Basel.
To the beginning of their2011–12 season season Dragović was member of the Basel team that won the2011 Uhrencup, beating bothHertha Berlin 3–0 andWest Ham United 2–1 to lead the table on goal difference aboveYoung Boys.[4] The team entered the2011–12 UEFA Champions League in the group stage. On 7 December 2011, during theChampions League group C match at home in theSt. Jakob-Park Dragović helped Basel defeatManchester United to make his team become the first Swiss club to advance to the knockout stage of a Champions League. Basel won the match 2–1, sending United out of the Champions league.[5] But in the round of 16, despite a 1–0 home win againstBayern Munich, it was the German team who advanced to the next round.
Dragović scored his first league goal for his new club on 1 October 2011 in the 3–0 home win againstServette FC.[6] At the end of the 2011–12 season he won theDouble with his new club. They won theLeague Championship title with 20 points advantage.[7] The team won theSwiss Cup, winning the final 4–2 in a penalty shootout againstLuzern.[8]
Basel had started in the2012–13 UEFA Champions League in the qualifying rounds. But were knocked out of the competition byCFR Cluj in the play-off round. They then continued in the2012–13 UEFA Europa League group stage. Ending the group in second position, Basel continued in the knockout phase. Dragović scored a goal in the second leg of quarter-final matches againstTottenham Hotspur, whom they beat 4–1 on penalties after a 4–4 aggregate draw to progress to the semi-finals.[9] In thesemi-finals, Basel were matched against the reigningUEFA Champions League holdersChelsea. Chelsea won both games advancing 5–2 on aggregate, eventually winning the competition.[10] Basel played a total of 20 European matches that season. Dragović missed solely the second leg of the semi-final atStamford Bridge due to a yellow card suspension, but had played in all the other European ties.
In the2012–13 domestic championship Dragović played in 32 of the 36 games and he scored three goals. In the away game in theStade de Genève on 3 March 2013 Dragović scored his team's first goal afterValentin Stocker played a pass from the bye-line backwards into the centre and Dragović slotted home from short range. AfterServette had equalised, Dragović headed home a corner, taken by team captainAlex Frei, to the 2–1 final score. Celebrating his goal, Dragović raised his arm in gesture. But refereeSascha Amhof understood this as rude gesture against the Servette fans and showed the scorer a yellow card. Because this was his second card in the game, Dragović was dismissed.[11] His third goal of the season was ten matchdays later in the home game in theSt. Jakob-Park as Basel won 2–0 against the same opponent, Servette.[12] At the end of the 2012–13 league season he won the Championship title with the team.[13] In the2012–13 Swiss Cup Basel reached the final, but were runners up behindGrasshopper Club, being defeated 4–3 on penalties, following a 1–1 draw after extra time.[14]
On 26 July 2013, it was announced that Dragović was leaving Basel and had signed forDynamo Kyiv.[15] During his time with the club Dragović played a total of 142 games for Basel scoring a total of nine goals. 77 of these games were in theSwiss Super League, eight in theSwiss Cup, 27 in the UEFA competitions (Champions League and Europa League) and 30 were friendly games. He scored four goals in the domestic league, one in the European games and the other was scored during the test games.[16]
On 26 July 2013, Dragović had signed a five-year contract with Ukrainian clubFC Dynamo Kyiv. On 17 May 2015, Dragović helped Dynamo beatDnipro 1–0 which was the game that won Dynamo their firstUkrainian Premier League title in six years.
On 4 November 2015, Dragović scored his first everUEFA Champions League goal in the group stage againstChelsea atStamford Bridge on the 78th minute, as well as scoring an own goal on the 34th. Dynamo ended up losing that match 2–1. In that season he helped Dynamo reach the last sixteen of theUEFA Champions League for the first time in 16 years, getting knocked out byManchester City 1–3 on aggregate.
Dragović also used to be part of the national under-17 and under-19 Austrian teams. He was called up forthe national team of Austria for their2010 FIFA World Cup qualification match againstRomania, but due to an injury he was unable to play. He was called up again by the Austrian head coachDietmar Constantini for the match againstSerbia. He made his international debut in this match on 6 June 2009. Dragović scored his first goal on 18 November 2014 in a 1–2 home defeat toBrazil.
He made the final squad forEuro 2016, where he was sent-off in the first game againstHungary,[20] and missed apenalty in the decisive match againstIceland, which ended in a 2–1 defeat.[21]
Dragović was part of the Austrian team which qualified to theEuro 2020 round of 16.[22] On 29 March 2022, he played his 100th match for Austria in a 2–2 draw againstScotland.[23]
In May 2012, Dragović attracted criticism in Swiss media for playfully slapping theSwiss defence and sport ministerUeli Maurer'sbald head duringFC Basel'sSwiss Cup win medal ceremony.[25] After making further comments about being reluctant to apologize—stating that "on the inside everybody knows, so I believe, that it was very, very much fun doing it"—Dragović reportedly traveled to theBundeshaus inBern in order to personally apologize to Maurer.[26]