Elezaj was born in Turin, Italy to Albanian parents originally from Vlorë. He started his youth career at Torino F.C. academies and at age of 6 he was invited for a trial at Juventus F.C. Youth Sector in 2003 which he successfully passed and was transferred there. On 1 July 2013, he did well at Primavera team and was included in the Juventus 2013–14 UEFA Youth League squad in which he served as a second choice behind Emil Audero. In January 2014, he was loaned out to S.S. Virtus Lanciano until the end of the 2014–15 season and played with Primavera side in total 11 matches.
Pro Vercelli
After return from loan, he was sold to F.C. Pro Vercelli for a fee of £375k. At Pro vercelli he became part of the first team, as he was trained with coach Cristiano Scazzola, but played as a first choice with primavera team in total 14 matches where he kept 3 clean-sheets. Elezaj participated with the first team in the 2015–16 Coppa Italia match against Alessandria on 9 August 2015, in which he was an unused substitute.
Loan to Sliema Wanderers
On 15 August 2015, Elezaj was loaned out to Maltese side Sliema Wanderers F.C. to gain more professional experience. He made it his debut on 21 August 2015, in the opening match against Tarxien Rainbows coming on as a substitute in the 74th minute in place of the midfielderPeter Xuereb after first choice goalkeeper Glenn Zammit was sent-off as he received a Red card. In the continuation, he had a poor playing space serving mostly as a second choice but also a third choice playing in just two occasions in October. Then in April would come his time as he played the full 90-minutes match against Pembroke Athleta on 2 April 2016, where he managed to keep the clean-sheet as his side won 2–0. During the month, he played in 5 out 6 matches, kept clean-sheets in 2 matches and helped his side to win all 5 matches in which he played, including also the 2015–16 Maltese FA Trophy - Quarter-final one against St. Andrews, where Sliema won 2–1 to qualify for the semi-finals. Into the semi-finals Sliema beat Birkirkara 1–0, where Elezaj played the full 90-minutes to help Sliema reach the grand final. In the final, Sliema played against Balzan, and Elezaj was in the starting line-up. Elezaj kept the clean-sheet for the full 120-minutes match, including extra-time, and then the match directed to Penalty shoot-out, in which Elezaj saved a penalty of opponent and his side won 5–4 to be declared the champions of the 2015–16 Maltese FA Trophy. Elezaj concluded the 2015–16 season with 8 league appearances, only 1 as a substitute and 3 domestic cup matches.
Perugia
On 30 July 2016 he signed with Serie B side A.C. Perugia Calcio a 2-years contract with renewing option. Elezaj made his professional debut on 1 December 2016 in the 2015–16 Coppa Italia match against Genoa under coach Christian Bucchi. The regular 90-minutes finished in the 2–2 draw and advanced in the Extra time where Genoa won 4–3. Despite conceding 4 goals in total of 120 minutes Elezaj was rated highly with 7 after making several saves against the Argentinian duo Giovanni Simeone & Lucas Ocampos.
His inspirations are famous fellow Goalkeepers, Gianluigi Buffon and Joe Hart where he had the chance to train with them, Buffon during the time that he was trained with the first team of Juventus and with Hart during the time he had a trial at English giants Manchester City.
