Hancock, the head of the insurer's global property-casualty business, succeeds Robert Benmosche, who will retire in September.
Benmosche will take up an advisory role at AIG, the company said.
"His success in revitalizing AIG's property casualty business; and his strong leadership and inclusive relationship skills position him perfectly to lead the company to an even more prosperous future," said Robert Miller, chairman of AIG's board of directors.
Retiring CEO Benmosche, who has been with AIG since 2009, is credited with steering the company through the financial crisis. During his tenure, AIG fully repaid the $182.3 billion in government support it received in 2008 to stave off bankruptcy.
Benmosche, who was diagnosed with cancer in late 2010, was CEO of MetLife Inc from 1998 to 2006.
With Benmosche at the helm, the insurer's stock has more than doubled to Tuesday's close of $55.01 on the New York Stock Exchange. AIG shares were lower in after-hours trading.
Benmosche will now take up an advisory role at AIG, the company said in a statement.
-By Reuters with CNBC