Gazidis is of Greek origin, but was born in Johannesbourg, South Africa. At the age of 4 moved to Manchester in the United Kingdom and attended independent Manchester Grammar School. Being an adopted Mancunian from a young age, he supported Manchester City. Later he attended St Edmund Hall at the University of Oxford, where he was twice awarded a blue playing football against Cambridge in 1984 and 1985. He graduated with a degree in law in 1986 and in 1992 he moved to the United States to work for Latham & Watkins.

In 1994, he joined the founding management team of Major League Soccer, becoming in 2001 its deputy commissioner. He oversaw MLS’s key strategic and business decisions and its marketing arm, Soccer United Marketing, of which he was president. He also helped promote the Mexican Football Federation and the CONCACAF Gold Cup.

In November 2008 Gazidis accepted the post of chief executive of Arsenal, a post he formally took up from 1 January 2009. He succeeded former managing director Keith Edelman and was expected to take over many of the roles formerly undertaken by ex-vice-chairman David Dein.

In 2017–18, he oversaw significant personnel changes at the top of the club in preparation for then manager Arsene Wenger’s departure, which was announced in April 2018. Gazidis spearheaded the search for a new Manager, eventually hiring Unai Emery in May.

On 18 September 2018, Gazidis announced that he would be leaving Arsenal to join A.C. Milan. He took up the role on 1 December 2018. During January 2019 transfer window, Gazidis vetoed the arrivals of Zlatan Ibrahimovic and Cesc Fabregas in Milan despite their free agent status, citing their age as the main reason. He also opted to terminate Gonzalo Higuain’s loan deal six months in advance. On the other hand, he approved the transfers of Krzysztof Piątek and Lucas Paquetá, both carried out by Leonardo. In his first season as the club’s CEO, the team finished fifth in Serie A, which became its best result in six years, yet failed to qualify for Champions League.