"More than ever, today, Americans, like the rest of humanity, need a functioning global system through which the world's peoples can face global challenges together. And in order to function, the system still cries out for far-sighted American leadership, in the Truman tradition," he said.
Annan steps down at the end of the month, to be succeeded by Ban Ki-Moon of South Korea.
During his two five-year terms as UN leader, Annan has tangled often with George W Bush's administration, particularly over the invasion of Iraq in March 2003, launched without consent from the UN Security Council.
Bush administration officials have argued Washington should use the UN only to serve its national interest.
Annan said "none of our global institutions can accomplish much when the US remains aloof. But when it is fully engaged, the sky's the limit".
An multilateral century
As Washington reviews its policies in Iraq, Annan has pushed for greater involvement by Syria and Iran, a more inclusive political system and greater human rights protections.
Truman, who ordered two atomic bombs dropped on Japan in 1945 making the US the sole power in history to use nuclear weapons, learned from that experience that security from then on "must be collective and indivisible", Annan said.
"You Americans did so much, in the last century, to build an effective multilateral system, with the United Nations at its heart. Do you need it less today, and does it need you less, than 60 years ago?
"When power, especially military force, is used, the world will consider it legitimate only when convinced that it is being used for the right purpose, for broadly shared aims, in accordance with broadly accepted norms."
Rights promotion
The US has historically been a leader in human rights, Annan said.
"When it appears to abandon its own ideals and objectives, its friends abroad are naturally troubled and confused," he said in an apparent reference to charges of abuse at US prisons in Guantanamo Bay, Cuba, and Iraq's Abu Ghraib.
Annan hit out at Washington's opposition to expansion of the 15-nation Security Council as part of a reform drive.
He said: "It is only through multilateral institutions that states can hold each other to account.
"And that makes it very important to organise those institutions in a fair and democratic way, giving the poor and the weak some influence over the actions of the rich and the strong."