Gazidis - as revealed by Sportsmail last month - is leaving to take up a similar role at AC Milan, news that was confirmed by the club this morning.
Gazidis is set to begin his new role on December 1, bringing his 10-year association with Arsenal to an end.
"I believe that, after 10 years, it is the right time for me to step aside to allow new leadership, energy and ideas to take the club forward into this exciting new era".
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.
"Given this platform, I know many will think this is a odd time to be leaving but I believe it is the right time for me and for the club", he added.
Arsenal offered him the role after Ivan Gazidis parted ways from the club to join former Champions League victor AC Milan.
Venkatesham had been chief commercial director since 2010, after working for the London 2012 Olympic organising committee.
The Italian side, now 13th in Serie A, announced that they were pleased Gazidis had accepted their offer.
Milan president Paolo Scaroni said last week that negotiations with with Gazidis "are ongoing", adding that "the process of selecting a chief executive must be done well and calmly".
The club achieved three FA Cup wins in Gazidis' time at the club, in 2014, 2015 and 2017.
"It has been a pleasure working with Ivan".
In July, Elliott Management took control of the Italian club after Chinese businessman Yonghong Li, who past year had bought AC Milan from former Italian Prime Minister Silvio Berlusconi, was unable to honor a repayment that was due to the United States fund.
His son, Josh Kroenke, is deputy chairman of KSE and praised the work of Gazidis during his tenure at Arsenal. "The club has been transformed and he has recruited talented people, such as Raul, Vinai and of course Unai Emery, who will take us to future success".