McMahon, 74, was chairman of the club from 1996 to 2003, an era of great success for the Bombers.
He is credited with the club's off-field success and in his last year as chairman, 2003, Essendon was debt-free and recorded an operating profit of more than $5 million.
McMahon assumed the role of chairman after 40 years as general manager and chief executive of Ansett Airlines.
He was born in Essendon, went to school in Essendon, was married in Essendon, lived in Essendon and had played football for Essendon. He was Essendon through and through.
In statement, current chairman Paul Little extended his deepest sympathies to the McMahon family.
"As a club we are extremely saddened by the passing of Graeme, an Essendon man through and through," Little said.
McMahon is survived by his wife Glenys and sons Brett and Brad.
Details of McMahon's funeral will be communicated in the coming days.