Both chambers of parliament have already approved the reform. However, because it requires changes to the constitution, it must also be put to a public vote.
Critics say the plans could undermine judicial independence.
The reform will pass if it receives more "Yes" than "No" votes.
The referendum is also seen as a test of public support for both the right and left ahead of the next parliamentary election, scheduled for late 2027.
More than 46 million Italians are eligible to vote. Results are expected by Monday evening.
The restructuring is one of the flagship initiatives of Meloni's coalition of three right-wing and conservative parties, which has governed Italy for three and a half years.
Italy's political right has long argued that large parts of the judiciary are aligned with the left.
The opposition, meanwhile, accuses Meloni of seeking to bring the justice system under political influence.
At the heart of the reform is a plan to separate the career paths of judges and public prosecutors.
It would also introduce new self-governing bodies for both groups, with parliament involved in appointing members - potentially giving politicians greater influence over personnel decisions.
Despite the political dispute, there is broad agreement that Italy's justice system requires reform, even as opinions differ on how to achieve it.
Legal proceedings often take significantly longer than in many other European countries, and only four-in-10 Italians trust the judiciary, according to one survey.
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