The 62-year-old Mazzarri became the second man to be sent packing this season by the ailing Italian champions.
Mazzarri took charge in November after Rudi Garcia was dismissed Napoli then fell to ninth in Serie A ahead of Wednesday's Champions League last 16 clash with Barcelona.
"I thank Walter Mazzarri, a friend of the De Laurentiis family and Napoli, for having helped the team in a tricky period," said Napoli owner Aurelio De Laurentiis on X.
De Laurentiis confirmed that Calzona has been hired, reportedly until the end of the season, and will be in charge for the first-leg visit of Barca.
"It's always painful to sack a friend," said De Laurentiis in a short interview with Sky Sport.
"But we need to consider that in Naples you have to give Napoli fans something extra. Now we're going to try to do that with Francesco Calzona."
Calzona guided Slovakia to Euro 2024 but has a huge job on his hands as he returns to Napoli, where he was an assistant under Maurizio Sarri and during Luciano Spalletti's trophy-less first season with the club.
The 55-year-old will reportedly keep his job with Slovakia while also trying to drag Napoli back up the table after the disappointing end to Mazzarri's brief second spell in charge of southern Italy's biggest club.
Mazzarri helped return Napoli to the upper echelons of Italian football between 2009 and 2013.
He guided Napoli to the 2012 Italian Cup, beating undefeated league champions Juventus in the final to win the club's first trophy in over two decades.
Mazzarri left in May 2013 after finishing second in Italy's top flight with a thrilling team which boasted Ezequiel Lavezzi, Marek Hamsik and Edinson Cavani in attack.
