Claver-Carone, who will succeed current President Luis Alberto Moreno, will take office on October 1 for a five-year term, Xinhua news agency quoted the IDB as saying in a statement on Saturday.
The IDB will have a non-Latin American President for the first time.
Since its establishment 61 years ago, all members have followed an unwritten tradition that only Latin Americans will run the bank.
The US has never before even nominated a candidate to lead it.
Argentina, Costa Rica, Chile and Mexico had insisted on postponing the election until March 2021, arguing that the pandemic prevents a face-to-face election as would be appropriate.
Argentina and Costa Rica had put forward alternative candidates in an effort to keep the bank in Latin American hands but both withdrew in the days before the vote after failing to receive adequate support.
Claver-Carone is currently deputy assistant to President Donald Trump as well as senior director for Western Hemisphere Affairs at the National Security Council.
He has previously served as US representative to the International Monetary Fund and senior advisor to the Under Secretary for International Affairs at the Treasury Department.
The controversial vote came after several former US officials earlier this month made efforts to persuade IDB shareholders to postpone the election until next year.
“Claver-Carone is a political appointee in the Trump White House whose controversial nomination would break a longstanding precedent that a Latin American serves as President of the IDB,” they wrote in a joint statement.
They argued that electing Claver-Carone to a five-year term only weeks before the US presidential election would “trigger an immediate institutional crisis at the IDB and put in jeopardy US support for the bank”.
Established in 1959, the IDB seeks to finance viable economic, social, and institutional development projects and promote regional trade integration in Latin American and the Caribbean.