Cook conceded he made an "error of judgement" in mistakenly sending an offensive email to the mother of Nedum Onuoha, the City defender, which prompted an internal investigation led by Khaldoon Al Mubarak, the City chairman.
The email - intended for City's football administrator Brian Marwood - appeared to mock Onuoha's mother, who was suffering from cancer.
The departure of Cook leaves a void in the corridors of power at the Etihad Stadium and an opening for Anderton.
He was headhunted by Sheikh Mansour bin Zayed, the chairman of Al Jazira and owner of City, for the position at Jazira and interviewed by Cook and an Abu Dhabi-based City executive.
Anderton took up his role at the Mohammed bin Zayed Stadium in January 2010, and has impressed his paymasters by attracting a record crowd of 28,164 for a domestic UAE game, and overseeing the club's league and cup double last season.
John Macbeath, a director, has been installed as City's temporary CEO, but Anderton could work in tandem with John Williams, the former Blackburn Rovers chairman who joined City's football operations team in July.
Keen to build their global brand, City may transfer Anderton, whose background is ostensibly in marketing, from Abu Dhabi to Manchester, particularly as the outgoing Cook played an instrumental key role in the club's commercial growth.
Anderton was based in the United States as the global brand ambassador for Coca-Cola and has held various executive positions in sport, piloting the commercial campaigns of the Scottish Rugby Union and the ATP World Tour in tennis. He was also CEO at Hearts, the Scottish club.
Al Mubarek paid tribute to the outgoing Cook. "Garry has made a remarkable contribution to Manchester City Football Club over the past three years.
"His judgement in this matter should in no way lead to his accomplishments being overlooked.
"On every level, the club is unrecognisable from the organisation which he inherited and our staff and supporter services, community outreach and commercial activity have seen unparalleled growth under his direction with yet more projects to be realised on the horizon.
"On behalf of Sheikh Mansour and the board, I would like to thank him for his energy and tireless commitment to serving all those connected with the football club.
"He will always be welcome at Manchester City."
* With agency