Matthew Nimetz said in a statement that the agreement, which remains to be ratified by Greece, paves the way for "a firmer basis for peace and security in the Balkans".
The so-called Prespa agreement seeks to end a 27-year-long dispute between Athens and Skopje over the name Macedonia.
North Atlantic Treaty Organisation leader Jens Stoltenberg said the alliance strongly supports the full implementation of the agreement, under which Greece will lift its objections to Macedonia joining North Atlantic Treaty Organisation and the EU.
Mr Nimetz has been the United Nations secretary-general's personal envoy on the naming dispute since 1999 and was also US President Bill Clinton's special envoy as mediator on the issue from March 1994 to September 1995.
On Friday, 81 out of 120 members of the parliament supported the amendments needed to rename Macedonia to the Republic of North Macedonia, which is expected to open the way for the country's accession to NATO and the European Union.
Athens and Skopje reached the agreement on renaming Macedonia in June.
The deal was hammered out by Zoran Zaev, the Macedonian prime minister, and his Greek counterpart Alexis Tsipras.
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Delays had marked the October vote that launched the procedure to change the constitution, also with a two-thirds majority.
The EU's Federica Mogherini said Friday that the Macedonian parliament's approval of the constitutional changes is "a crucial step" in implementing the deal with Greece.
Under the agreement, after Skopje cements the name change with constitutional amendments, Tsipras is to push the ratification of the agreement through the Greek parliament.
For the Greeks, Macedonia evokes national pride as the cradle of Alexander the Great's ancient empire, a heritage that they guard jealously. In September, the move was supported by over 91 percent of Macedonians at a referendum.
Officials of the main opposition conservative VMRO-DPMNE party, who refused to take part in the parliamentary debate, denounced the name change as treason.
"The prime minister congratulated Mr Zaev on the successful conclusion of the process to revise the constitution of the former Yugoslav republic of Macedonia", Tsipras' office said in a statement.
Further complicating matters, Greek coalition partner Defence Minister Panos Kammenos, the leader of a smaller party, opposes the deal and has threatened to resign.
NATO Secretary General Jens Stoltenberg congratulated Zaev on Twitter and reiterated the Alliance's support for the deal with Athens.