The ex-Emirates supremo so closely linked to Arsene Wenger takes charge on a deal which runs until June 2020.
Gazidis' position was confirmed at a board meeting today.
The South Africa-born businessman, 54, spent his first day at the club last Saturday.
It was then he met manager Gennaro Gattuso and the team.
Much like his near ten-year tenure at Arsenal, Gazidis will be responsible for looking after the club's finances.
Calciomercato say the model Gazidis aims to implement at the Serie A club will be the same he used at Arsenal.
They claim this means "more than winning, therefore, it will be essential to be permanently among the top three/four in the standings to consistently qualify for the Champions League".
Milan currently sit fourth in Serie A, some 15 points behind leaders Juventus.
Reports in the Italian press say Gazidis is planning a big January transfer window.
Gunners outcast Aaron Ramsey is rumoured to be heading there when his Arsenal contract expires.
Flamengo's Lucas Paqueta has already signed and Gazidis is planning two more additions to the squad.
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