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.
The court said it was easy for Facebook to automate this process.
A Green lawmaker, Dieter Brosz, said Facebook could no longer claim it was just a platform and needed to take responsibility for tackling hate postings.
Internet giants including Facebook, Twitter and Google have all come under fire in many countries for failing to remove hate speech from their platforms promptly.
Last month, German ministers approved plans to fine social media firms up to 50m euros ($53.3m; £42.7m) if they fail to remove hate speech and fake news quickly.
The companies have recently announced measures to address the issue:
- Facebook said it would hire 3,000 people to help stop hate speech, child abuse and self-harm being broadcast on the website
- Google said changes on how its core search engine works would help stop the spread of fake news and hate speech