Outgoing President Joe Biden is scheduled to visit Rome this week.
Meloni thanked “everyone who helped bring Cecilia back.”
Sala’s partner, fellow journalist Daniele Raineri, told the Ansa news agency: “I spoke to her and she told me: ‘See you soon’, she was emotional and happy.”
Mohammad Abedini is due to stand trial in Milan on January 15, and Tehran has played down any link between the two cases in recent days.
Italian foreign intelligence chief Giovanni Caravelli is said to have personally traveled to Tehran to bring Sala back to Italy.
Her father, Renato Sala, told Ansa he was proud of her and praised the government for its “exceptional work”.
He said he had the impression that the situation had turned into “a game of chess, but with more than two players.”
Renato Sala also thanked Foreign Minister Antonio Tajani, noting that they had lived together for 12 years and had become friends.