Macedonia’s parliament approved a constitutional revision bringing a decades-old dispute with neighboring Greece one step closer to a resolution that would unblock the country’s membership to the West’s most powerful clubs.

The vote took place late Friday behind closed doors after hours of delay amid high tensions. It came on the heels of accusations—leveled by the deal’s supporters and opponents—of threats, blackmail and bribery. In all, 80 members of parliament backed the deal, which changes the name of the Balkan nation...