The two leaders met away from the media for more than an hour in central Rome, after which Macron vowed in a post on Twitter to work together "with dialogue and ambition".
Meloni's post-fascist Brothers of Italy won a historic victory in general elections on September 25.
Her new government is the most far-right in Italy since World War II, and takes power at a time of soaring inflation and an energy crisis linked to Russia's invasion of Ukraine.
But there are already tensions within her coalition, which includes Matteo Salvini's far-right League and former premier Silvio Berlusconi's right-wing Forza Italia.
Meloni was forced this week to repeat her support for Ukraine and Western sanctions against Russia after Berlusconi was recorded defending President Vladimir Putin.
"We must be united, there are emergencies the country is facing. We have to work together," the 45-year-old told her first cabinet meeting Sunday.
The prospect of a Eurosceptic, populist government leading the eurozone's third largest economy has sparked concern among Italy's allies, particularly in the European Union.
European Commission President Ursula von der Leyen said she had a "good first call" Saturday with Meloni, saying she looked forward to "constructive cooperation".
Meloni's office described Sunday's talks with Macron as informal but "friendly and fruitful", covering the need for urgent and common action on high energy bills, support for Ukraine and migration.
"Meloni and Macron agreed on the willingness to continue to cooperate on the major common challenges at European level and in respect of mutual national interests," it said in a statement.
Macron's aides had refused to confirm the meeting with Meloni until the last minute, and it came during a pre-planned trip to Rome and the Vatican.
Before his tweet about their meeting -- including a photo of the pair on a rooftop at dusk -- Macron posted a picture of outgoing prime minister Mario Draghi, with the words "Grazie (thank you) Mario".
After Meloni and her 24 ministers were sworn in on Saturday before President Sergio Mattarella, she declared her intention to get "straight to work".
On Sunday she joined Draghi, a former European Central Bank chief in office since February 2021, for a symbolic handover of power.
They held private talks for almost 90 minutes before a smiling Draghi gave Meloni a small bell used in cabinet debates, which she, grinning, rang a few times for the television cameras.
As a teenage activist, Meloni praised late dictator Benito Mussolini, but insists fascism is history and has transformed her once marginal party into a national force.
During 18 months as the only real opposition to Draghi's national unity government, Meloni swept up disillusioned voters and secured 26 percent of the vote last month.
She presents herself as a straight-talking "Christian mother", a defender of traditional values and of Italy's national interests, but her party has never held power.
Meloni has sought to reassure investors that Italy's debt-laden economy is safe in her hands, and has appointed Giancarlo Giorgetti as economy minister.
Giorgetti, who served as minister of economic development under Draghi, is considered one of the more moderate, pro-Europe members of Salvini's League.
Draghi's energy minister, Roberto Cingolani, will stay on as government advisor as Italy tries to wean itself off Russian gas, reports said.
Pope Francis had earlier marked the start of the new government in his weekly Angelus Sunday, offering his prayers for "unity and peace in Italy".
Meloni's party no longer wants Italy to leave the EU's single currency but remains strongly Eurosceptic, as is the League, which won nine percent in the elections.
However, she named as foreign minister and deputy prime minister Antonio Tajani, a former president of the European Parliament who co-founded Forza Italia with Berlusconi. Their party won eight percent of the vote last month.
Salvini will serve as deputy prime minister and minister of infrastructure and transport.
Like Berlusconi, Salvini is a long-time fan of Putin and has criticised Western sanctions imposed on Russia over its invasion of Ukraine.
The tensions with her allies reinforce doubts as to how long Meloni can keep her coalition together, in a country that has had almost 70 governments since 1946.
On Saturday she again affirmed her desire to work with NATO, "a bulwark of common values we'll never stop standing for".
NATO chief Jens Stoltenberg and US President Joe Biden sent their congratulations, as did Ukrainian leader Volodymyr Zelensky.