In a statement on Thursday, the New York-based company said that Hancock had informed the board of his intention to resign. The company will conduct a comprehensive search for a successor to Hancock. "Without wholehearted shareholder support for my continued leadership, a protracted period of uncertainty could undermine the progress we have made and damage the interests of our policyholders, employees, regulators, debtholders, and shareholders".
AIG's shares were up 0.9 percent at $64 in early trading. Icahn applauded the board's decision in a Twitter post on Thursday.
Keefe, Bruyette & Woods analyst Meyer Shields called the move a "significant positive" for the stock, according to Bloomberg.
The 58-year-old's successor will be the firm's seventh CEO in seven years.
Hancock is AIG's fifth chief executive since Maurice "Hank" Greenberg was ousted in 2005.
American International Group disclosed that Hancock was entrusted with the responsibility of the current position in September 2014 on his selection by its board.
Mr. Icahn quieted his criticism after he won representation on AIG's board, the insurer slashed its management ranks and proposed a turnaround plan that included shrinking its US casualty book and selling certain units - but falling short of a full breakup.
In January, AIG announced a deal to pay $9.8 billion to National Indemnity Co, a unit of Berkshire Hathaway Inc to cover most of AIG's US commercial long-tail exposures, or liabilities for claims that take a long time to be settled, for accidents that occurred in 2015 and prior. The CEO has said that a downgrade could jeopardize relationships with some customers.
AIG board members were frustrated by the fourth-quarter results, which included the second surprise charge in a row, people familiar with the matter said.
AIG has endured more than a decade of management turmoil.
The late Robert Benmosche ran AIG for more than five years, starting in 2009. The insurer repaid the government rescue in 2012.