He started his youth career at the age of 8 at Liria Prizren school academy Getoari, from where he moved to the youth teams of the club in 2008 where he stayed for 3 years until 2011, before moving to the most successful football club in Kosovo, FC Prishtina, at the age of 16. While at Prishtina, Gavazaj along with his teammates Liridon Latifi and Lorik Aliu went on trial with German giants Borussia Dortmund which initially lasted a week, but was then extended so they players could feature in the Blue Stars FIFA Youth Cup and the RUHR-Cup International youth tournament. Three players were not offered contracts at the end of their trials and subsequently returned to FC Prishtina. Standing for two years at FC Prishtina, on 8 August 2013 he signed for Belgium side K.A.A. Gent in the Belgian Pro League.
In December 2013 Gavazaj gained entry with the first team of Gent, as he called up for the match against Anderlecht on 15 December 2013, finished in the loss 1–2, in where Gavazaj was an unused substitute for the entire match. He made it his first debut for Gent at age 18 against Genk on 26 January 2014, coming on as a substitute in the stoppage time 90'+2 in place of Yaya Soumahoro, where the match finished in the away victory 1–2. The second match for Gent came on 8 February 2014 against Oostende, where he came in as a substitute in the 81st minute in the place of Brecht Dejaegere. After some games as an unused substitute, Gavazaj played on 1 March 2014 against Sporting Charleroi, where he included in the starting line up and got substituted off in the 57th minute for Renato Neto. Gavazaj played on 8 March 2014 in a match against Cercle Brugge, coming on as a substitute in the 73rd minute of the match in place of Nana Akwasi Asare and in the 81st minute Gavazaj gave an assist for the 3rd goal of the match scored by Mahamadou Habibou, were the match finished as an away victory 1–4. After that game, he included in best eleven of the week in Belgian Pro League. Gavazaj played on 3 May 2014 against Lierse, coming on as a substitute in the 61st minute in place of Danijel Milićević and 13 minutes after in the 74th' he assisted the latest match's goal scored by Hervé Kage, as the match finished in the victory 4–0, with 3 other goals scored by Habib Habibou as a Hat-trick.
Gavazaj did not play for 4 months since the start of the 2014–15 season and this due to several injuries that have followed him, including and at one call-up with Albania national under-21 football team. But anyway in February 2015 he declared for a Kosovo-Albanian newspaper, Gazeta Olle, that he was healthy and got returned training with the team as a fully recovered.
On 6 July 2015 Gavazaj was loaned to Belgian Second Division side Roeselare to gain more experience by enjoying more playing options. However, during the first half of the 2015–16 Belgian Second Division season Gavazaj featured in just 5 matches playing in all as a second-half substitute.
Following loan return from Roeselare, Gavazaj was sold to Skënderbeu Korçë singing on 26 June 2016 prior the start of the Summer Transfers Window.
Gavazaj was member of the Kosovo national youth football teams for 2 years before moving to the Albania national youth football teams principally at the Albania national under-19 football team.
He was included in Foto Strakosha's U19 squad for their 2013 UEFA European Under-19 Championship qualification. In the first match of the tournament on 12 October 2012 against Italy U19, Gavazaj made it his debut by playing as a starter and got substituted off in the 79th minute for Lorenc Shehaj in a match finished as a 3–0 loss. In the next match against Belgium U19 he included in the starting line-up and after 26 minutes of play, he scored a long-rage goal away from 33 metters. The match finished in the 3–1 victory.
After his good performances with under-19 side Gavazaj was called up at Albania national under-21 football team by coach Skënder Gega and participated in three 2015 UEFA European Under-21 Championship qualification Group 4 matches in months August, September and October 2013 before returning again to Albania U19 to participate in another qualification campaign for Under-19s, the 2014 UEFA European Under-19 Championship qualifying round. He played all 3 matches as full 90 minutes against Greece U19 on 12 November 2013 in the 1–0 loss and two 1–1 draws days later against Slovakia U19 & Bulgaria U19. Albania U19 finished in the last place in the ranking table.
Gavazaj was called up at Albania national under-20 football team by coach Skënder Gega to participate in the 2013 Mediterranean Games football tournament which began on 19 June 2013 in Mersin, Turkey. Gavazaj played in all 5 Albania U20s matches in which 2 of them were full 90-minutes, 1 as starter until second half and 2 others as a substitute. Albania U20 was ranked in the last place out of 8 teams.
Following his good performances with Albania U19, Gavazaj received his first call-up at Albania national under-21 football team by coach Skënder Gega for the match against Austria, valid for the 2015 UEFA European Under-21 Championship qualification Group 4. He made his debut against Austria on 14 August 2013, by coming on as a substitute in the 65th minute in place of Herolind Shala and the match finished in the loss 0–1. He made it another two appearances in the 2015 UEFA European Under-21 Championship qualification Group 4, playing against Spain on 9 September 2013, finished in the loss 4–0 and against Bosnia and Herzegovina on 15 October 2013, finished in the loss 0–1.
Gavazaj called up again in Albania U21 for the closing match of the 2015 UEFA European Under-21 Championship qualification Group 4 on 5 March 2014, against Austria, which previously he were called up for the first time and had made it his debut for Albania U21. He played against Austria U21 as a starter and scored the match opening goal in the 19th minute, where the match finished in away 1–3 victory with other goals scored by Vasil Shkurti and Herolind Shala.
Gavazaj was called up for the next tournament even of Albania U21, the 2017 UEFA European Under-21 Championship qualification opening match against Liechtenstein on 28 March 2015. Under the same coach Skënder Gega he retained his starting place against Liechtenstein where in the 31st minute he managed to take a personal action from the left side and to provide an assist for Liridon Latifi, who inside the area scores the opening goal. Then in the second-half Rey Manaj which came in as a substitute, scored the second goal to give Albania U21 the 2–0 victory.
Gavazaj was called up by a new appointed coach at the Albania U21, Redi Jupi, for the 2017 UEFA European Under-21 Championship qualification matches against Israel & Portugal on 3 & 8 September 2015. For the match against Israel, he was named as the national side Captain playing the full 90-minutes match finished in the 1–1 draw.As of 14 September 2017
As of 12 October 2012
. Albania U19 score listed first, score column indicates score after each Gavazaj's goal.
As of 5 March 2014
. Albania U21 score listed first, score column indicates score after each Gavazaj's goal.