A product of the famous HNK Hajduk Splityouth academy and one-time Croatian U16, U17 and U19 national team player, Goran began his senior career in 2004 on loan at HNK Trogir, then playing in the 1. Županijska liga Splitsko-dalmatinska. The club topped the table and moved to the 3. HNL – South and Goran played another season there. Dismissed from Hajduk as surplus, Goran played two more seasons in 3. HNL – South, first for NK Zagora Unešić and then NK Junak Sinj, where he was the star player in a season that resulted in their promotion to 2. HNL, catching the attention of some 1. HNL clubs.
In 2008, Blažević moved to HNK Šibenik, quickly becoming a first team player. With HNK Šibenik in financial difficulties, Blažević cancelled his contract with the club in December 2011.
In January 2012, Blažević signed a four-year contract with Hajduk Split. As Hajduk's former first choice goalkeeper Subašić had just left the club for French Ligue 2 side AS Monaco, Blažević immediately was bestowed the first choice at the historic club.
In September 2013 Blažević passed successfully his medical and signed a 2-year contract with Bulgarian side Levski Sofia. He quickly became first choice at the expense of Plamen Iliev, but gradually lost his starting spot following the appointment of Antoni Zdravkov.
Torpedo Moscow
In the summer of 2014, Blažević moved to the Russianside FC Torpedo Moscow, where he took on the role of backup to Yuri Zhevnov. He went on to play only one game for the club, in the Russian Cup, a 2-0 away loss to CSKA Moscow, before succumbing to an Achilles tendon injury requiring surgery at the start of 2015. Blažević was released by the club that summer.
At the start of 2016, Blažević signed to Slaven Belupo in Croatia, playing only two matches before a shoulder injury forced him off the pitch until the end of the season. Returning in the summer of 2016. At the start of the season, Blažević, not fully healed, lost the starting place to Antun Marković, but after full recovery, took it over in September 2016. Marković would ultimately secure said position the following year, and Blažević would stay second-choice until his contract expired at the end of 2017.
Return to Hajduk Split
In September 2018, Blažević was signed by his former club HNK Hajduk Split, requiring a backup goalkeeper for Josip Posavec and Marin Ljubić, due to Tomislav Duka's injury. In October 2018, after Posavec was injured and Ljubić relegated to the reserve team, Blažević stepped in, featuring in 5 league matches and one cup match. Following Posavec and Duka's recovery in December 2018, Blažević was relegated again to a backup role, before he was released from the club after his second one-season contract expired at the end of June 2020.
