AHI Holding, one of the investment bank's real estate funds, is expected to finalize the deal for the company's 23-floor headquarters today.
Excluding two basement floors, the building has 1.43 million square feet of office space.
The sale represents the highest price paid for a building of any kind in South Korea, according to a report from the Yonhap news agency.
Morgan Stanley had been vying for the building with other bidders including Kookmin Bank of Korea and Australia's Macquarie Bank.
News of the deal follows Morgan Stanley recently agreeing to acquire Investa Property Group, paying $4 billion for the largest holder of office buildings in Australia.
Meanwhile, the US bank also paid $2.9 billion for Crescent Real Estate Equities.
A regulatory filing from Daewoo confirmed that it will use the proceeds from the sale to "repay debt and invest in assets with a high-return outlook".