He grew up in Södertälje and started playing football with local club Assyriska FF as a youngster, before joining the academy of IF Brommapojkarna in 2005.

[1][2][3][4] In 2010, Blažević was promoted to Brommapojkarna's first team in Allsvenskan, acting as a back-up to Kristoffer Nordfeldt when the club was relegated from the top tier.

[5][9] In April 2016, Blažević was close to joining Jönköpings Södra in Superettan due to an emergency crisis, but the deal fell through after the Swedish Football Association did not grant an exemption outside of the transfer window.

[15] In 2019, Blažević mainly served as a backup to both Gianluca Curci and Johan Wiland, but still ended the season making 13 league appearances in total.

[20][21] Blažević featured in the final of the 2021–22 Svenska Cupen, in which Hammarby lost by 4–5 on penalties to Malmö FF after the game ended in a 0–0 draw.