The order, requested by a Jewish student union and rights groups, concerned anti-Semitic material but could open the floodgates to legal pursuit of Twitter users who post a wide range of messages deemed illegal or offensive.
"This is an excellent decision, which we hope will bring an end to the feeling of impunity that fuels the worst excesses," said Stephane Lilti, lawyer for the groups who sought the ruling.
The anti-Semitic messages started appearing last October, and have since been deleted.
The Paris court gave privately-held Twitter, whose general policy is that it does not control content posted on its network, 15 days to hand over data identifying people who have published messages judged anti-Semitic.
The court also ordered Twitter to set up a system in France that helps people draw attention to illegal content. Under French law, people found guilty of inciting racial hatred can be jailed for a year and fined.
Twitter's lawyer in France, Alexandra Neri, declined to comment.
Failure to comply would expose the firm, founded in 2006 and now boasting 140 million monthly active users worldwide, to daily fines of €1,000 if the groups who sought the order request it, which Lilti said they would not hesitate to do.
A rights group involved in the case was quick to point out that the injunction, while limited to a case of anti-Semitic traffic, set a precedent that could also have a wider impact.
"This marks a decisive step forward in the battle against racist, homophobic and anti-Semitic offences on the Internet," the International League Against Racism and Anti-Semitism (LICRA) said in a statement.
"Nobody can ignore French law, not even the giants of the American digital economy."
For a first time, Twitter deployed a new message-blocker in Germany last October to jam the posting of messages by a neo-Nazi group banned by police.
A tool Twitter calls "country withheld content" allows it to censor tweets considered illegal in a given country.