Donald Rumsfeld, 74, resigned Wednesday after Democrats won a narrow majority in the Senate and took total control of Congress. President George W. Bush has proposed former CIA Director Robert Gates as the new defense secretary.
"I hope for further constructive development of relations between the Russian and U.S. defense ministries," said Sergei Ivanov, who is also a deputy prime minister.
Ivanov said both ministries should defend national interests, and rise to challenges and threats facing Russia and the United States.
He said Rumsfeld's resignation was linked to domestic developments in the U.S., but gave no political assessments of Rumsfeld and his likely successor.
Ivanov said he had enjoyed good relations with Rumsfeld, but that each of them had "first and foremost defended the interests of ministry and state."