Sanllehi is now the head of football relations, and will become the head of football going forward, with Venkatesham moving from his chief commercial officer role to managing director.
Following the latter, the club agreed a two-year deal to retain Wenger at the club, although the Frenchman departed his role as manager in 2018.
Raul Sanllehi, who arrived from Barcelona last season, and Vinai Venkatesham will effectively replace Gazidis - who has earned £20.9million since arriving at Arsenal in 2009 -at the helm.
"Every conversation I have with him is speaking about us and about our work here every day", he said. This will remain central to the club's philosophy.
"I am now looking forward to joining one of the world's other great clubs, AC Milan, and working to restore it to its rightful place in football".
Venkatesham said: "From the day I joined Arsenal I have always considered it a privilege to serve this great football club, so it's an honour to take up the position of managing director and lead the club alongside Raul".
Chrissy Teigen says we've been saying her name all wrong
It's Tie-gen." In the background, her mom hilariously affirms the shocking revelation with a "Yep!" The reality? It's TIE-GEN. Taxi going to the wrong airport? 'I don't correct people, ever. "I'll change my flight".
Milan, meanwhile, released a statement confirming Gazidis would begin his tenure at the club from December 1.
Deputy chairman of Arsenal's majority owner Kroenke Sports and Entertainment UK Josh Kroenke added: "It has been a pleasure working with Ivan". I believe in the positive force of change, both for me and for the club.
AC Milan's executive chairman, Scaroni, was quick to welcome Gazidis to the football club and went on to say that the appointment only bodes well for the future.
'Thank you Ivan for helping me to be part of the Arsenal, ' he wrote on Twitter.
'We will miss the wisdom, insight and energy he brought to our club on a daily basis over the past decade.
Elliot is reportedly intent on transforming Milan's fortunes before it attempts to sell the club on for a profit, which could see Gazidis earn a huge windfall from the stake he would assume should he head to Italy.