The case was brought by the country's Green Party after its leader was targeted by a false account.
The court said postings not just in Austria but worldwide must be deleted. Facebook has not yet commented.
The ruling is seen as a victory for campaigners who want to make social media platforms combat online trolling.
The appeals court in Vienna ruled that postings against Greens' leader Eva Glawischnig as any verbatim repostings should be removed.
It added that merely blocking the messages in Austria without removing them for users abroad was not sufficient.