"Grandissimo," Meloni said when Sinner walked into her office in the Chigi Palace.
"I want to show you something," Meloni then said, before turning on a replay of match point from Sinner's five-set victory over Daniil Medvedev in Sunday's final, during which the Italian rallied from two sets down.
"It's nice to experience it after the fact," Sinner said.
Sinner and Meloni posed for photos as they held aloft together first the Australian Open trophy and then the Italian flag. Sinner then gave Meloni a warm embrace to end the meeting.
Sinner had arrived at Rome's airport also holding a green bag with gold-coloured straps that, to judge from the outline of its contents, contained his Australian Open trophy.
Jannik Sinner's image was projected onto Turin's Mole Antonelliana after his Australian Open win. (EPA PHOTO)
Sinner is the first Italian man to win a Grand Slam singles title in nearly a half century — since Adriano Panatta won the French Open trophy in 1976.
In November, Sinner also led Italy to their first Davis Cup title since 1976.
Sinner will hold a news conference with Italian Tennis and Padel Federation president Angelo Binaghi on Wednesday. He and his Davis Cup teammates are scheduled to be honoured by Italian President Sergio Mattarella on Thursday.
On Monday, Pope Francis congratulated Italy for Sinner's victory.
