"At the height of the Cold War, he championed and negotiated landmark agreements on human rights and nuclear arms reductions," Biden said in a statement issued by the White House.
"Fittingly, Max devoted his final years to passionately advancing the vision he shared with Ronald Reagan and others -- that of a world without nuclear weapons," the statement added.
Kampelman died Friday at his home in the capital, The New York Times reported.
He was 92.
His son, Jeffrey, said Kampelman suffered from congestive heart failure.
A Washington lawyer who negotiated with the Soviet Union for two U.S. presidents, Kampelman was born in New York and grew up in the city. He graduated from New York University and its law school.
Kampelman was a conscientious objector in World War II and was sent to Minneapolis to be a subject in a study of starvation. He became an aide to Hubert Humphrey when he was mayor of Minneapolis and moved to Washington with him when he was elected to the Senate.
President Jimmy Carter asked Kampelman to head a team in Madrid negotiating a human-rights agreement with the Soviet Union. Kampelman, who had been told the effort would take three months, spent three years negotiating the agreement that was signed in 1983, when Ronald Reagan had become president.
Two years later, after Kampelman had served as an adviser to Walter Mondale's presidential campaign, Reagan invited him to lead arms-reduction talks with the Soviet Union in Geneva, Switzerland.
"We cannot wish it away," Kampelman said of the Soviet Union as the talks began in March 1985. "It is here and it is militarily powerful. We share the same globe. We must try to find a formula under which we can live together in dignity. We must engage in that pursuit of peace without illusion but with persistence, regardless of provocation."
Kampelman returned to Washington two years before the Strategic Arms Reduction Treaty was signed in 1991.
In Washington, Kampelman also was chairman of WETA, the capital's public television station, and founded and moderated the program "Washington Week in Review."