The New York-based insurer said that Benmosche was expected to also resign from the board and take on an advisory role with the company.
Benmosche, 70, became CEO in August 2009. He led AIG's effort to repay its $182 billion bailout from the federal government at the height of the 2008 financial crisis. AIG has also been restructuring and selling off businesses, cutting its size in half.
He had said that he planned to step down in early 2015. The company had said the board was considering internal and external candidates to replace him.
Hancock's experience in financial services, including risk management, and his success in revitalizing the property casualty business were among factors in his favor, said Chairman Robert Miller in a statement Tuesday.
Shares of AIG fell 6 cents to close at $55.01. In late trading after the CEO announcement, they were down 11 cents to $54.90. The stock has gained 8 percent this year.