Raised in a football-loving family, Alagich began playing football in Valley View, South Australia, with Valley View SC and Port Adelaide Pirates in their under-6.^[4] He also spent time at Adelaide Croatia Raiders, where he played for nearly six years, before being signed by Adelaide United in 2019.^[4]

For Adelaide United Youth, Alagich would make 22 league appearances, including one goal in the National Premier Leagues South Australia.^[4] He made his first-team debut as a substitute on 26 September 2021 in a 3–1 extra-time cup victory over Floreat Athena.^[5][6] Alagich was later added to the first-team squad after signing a scholarship deal with the club in January 2022.^[4]

Alagich made his starting debut on 11 November 2022 in a 3–0 victory over rivals Melbourne Victory at Coopers Stadium.^[7][8][9] He and Richie became the third father-son duo to playe for the club in a competitive match.^[10] Alagich was named Adelaide United's A-League Men Rising Star, jointly with Nestor Irankunda, and extended on a two-year deal with the club in July 2023.^[11][12]

Raised in South Australia, Alagich is the son of former Australian professional footballer Richie Alagich, who played over 400 competitive matches, including 92 for Adelaide United.^[13] His grandfather, Colin Alagich Snr, was a footballer and referee whilst his aunty, Dianne Alagich, was also a professional footballer.^[14] Alagich is of distant Croatian descent - As his great, great-uncle migrated to Australia from Yugoslavia in 1932.^[14]