Greece only agreed to stop blocking them if Macedonia formally renamed itself to The Republic of North Macedonia. Lawmakers in both countries agreed the deal last year, which is due to take effect soon.
Greece objected to Macedonia's name, which it adopted in 1991 when Yugoslavia collapsed, because Macedonia is also the name of a region of Greece.
Politicians in Greece argued that the name "Macedonia" suggested that the country had ambitions to one day rule Greek Macedonia as well.
Any country could technically veto the process. But previously the only country to object was Greece, which has now changed its stance because of the name change.
The deal between the two countries, known as the Prespa Agreement, was signed in June 2018.
If the other 29 states ratify the approval, Macedonia will then pass its own legislation, at which point it will become a full NATO member.
Their decision to change their name split the country. Macedonians rejected one attempt to change the name in an advisory referendum held in October 2018. The country's parliament later agreed the change anyway.
NATO Secretary General Jens Stoltenberg described Wednesday as "a historic day."
The last country to join NATO was Montenegro in 2017. Bosnia and Herzegovina, Georgia, and Ukraine have all expressed interest in joining.
They are also expected to meet certain political, economic and military criteria, including spending a minimum proportion of GDP on their militaries.