The saga began when Strauss-Kahn, the former managing director, was arrested at Kennedy International Airport in New York on May 15 for allegedly sexually assaulting a chambermaid at a hotel.
Three days later, Strauss-Kahn, once a favorite to challenge Nicolas Sarkozy for the presidency of France, resigned from his position as leader of the influential monetary fund.
The U.S. Treasury Department Tuesday threw its support behind Lagarde, with Treasury Secretary Timothy Geithner calling her an "exceptional talent."
By then, Largade, who will become the first woman to serve as the head of the IMF, had garnered support from most of Europe, China and Russia.
Lagarde has been closely involved with issues concerning bailouts of Greece, Ireland and Portugal, which added leverage to her candidacy.
Russian Finance Minister Alexei Kudrin said, "I think that she has all the necessary qualities and when the (IMF) board of directors votes, we will support her candidacy."
Lagarde's closest rival for the position, Agustin Carstens, the governor of the central bank of Mexico, had support from countries represented by only 12 percent of the IMF's governing board, The Wall Street Journal reported.
Carstens had support from much of Latin America, Canada, Australia and Mexico, but some of that support was considered symbolic, the Journal said.
The IMF board said Lagarde was offered a five-year term to serve as managing director and madame chairman of the executive board.
The fund serves 187 member countries and has a staff of 2,700 in 140 offices around the world.