The ruling follows a legal complaint lodged in October by France's Union of Jewish Students (UEJF) which argued that numerous tweets had breached French law prohibiting incitement to racial hatred.
The union had been pressing Twitter to exercise tighter control of what appeared on its site following a deluge of anti-Semitic messages posted under the hashtag #unbonjuif (#agoodjew).
Twitter later removed some of the offending tweets.
The UEJF took legal action aimed at forcing Twitter to identify the authors of some of the posts.
The court in Paris on Thursday said the site must do this "within the framework of its French site".
In October, Twitter suspended the account of a neo-Nazi group in Germany following a request from the government in Berlin. It was the first time the US firm had acted on a request of that nature from a government.