Empoli, who are battling relegation, opened the scoring after four minutes when striker Alberto Cerri headed in a close-range cross for what proved to be the winner.
"We're not mathematically out of it but the chances of us getting there are very slim," said Calzona after his first league defeat as Napoli boss
Italian club Napoli said they were "stunned" after Italy's football federation (FIGC) on Tuesday dropped a probe into alleged racist comments made by Inter Milan's Francesco Acerbi to Napoli's Juan Jesus due to lack of evidence.
Barcelona came flying out of the blocks with goals from Fermin Lopez and Joao Cancelo but after Amir Rrahmani pulled one back, the Catalans started to sweat.
Nigeria forward Osimhen rolled home the leveller with 15 minutes left of the first leg of their Champions League last 16 tie at the Stadio Diego Armando Maradona, in his first match since returning from the Africa Cup of Nations.
The 62-year-old Mazzarri became the second man to be sent packing this season by the ailing Italian champions.
Nigeria striker Victor Osimhen will leave Napoli this summer and join "either Real Madrid, Paris Saint-Germain or an English club", the Italian club's president and owner Aurelio De Laurentiis said on Friday.
Napoli put aside their recent troubles on Thursday by reaching the Italian Super Cup final with a convincing 3-0 win over Fiorentina in Riyadh.
Napoli striker Victor Osimhen has replied angrily to suggestions that he would leave the Serie A side for Saudi Arabia at the end of this season from agent Mamuka Jugeli, who represents club team mate Khvicha Kvaratskhelia.
Defending Serie A champions Napoli saw their non-winning league ways continue Friday with a goalless draw against mid-table Monza.
Yann Bissek's first goal in Italy just before half-time at the San Siro and a late finish from Nicolo Barella were enough for Inter to bounce back from cup defeat to Bologna midweek and keep closest challengers Juventus at bay.
Victor Osimhen has signed a contract extension with Napoli until 2026, the Serie A champions announced on Saturday.
The Italian champions qualified second in Group C behind Real Madrid thanks to Serder Saatci's clumsy own goal and Victor Osimhen's first strike since early October, both which came in the first half.
Osimhen, who led Napoli to win the Italian League title last season, beat Paris Saint-Germain's Morocco right-back, Achraf Hakimi and Liverpool's Egypt winger, Mohamed Salah, to win the top award.
Real Madrid romped to a 4-2 win over Napoli on Wednesday to qualify in first place from their Champions League group.
Mazzarri's team are still seven points behind leaders Inter Milan who travel to Turin on Sunday for the weekend's big match at second-placed Juventus
Inter Milan and Juventus face off on Sunday in a clash between Serie A's two title favourites and historic rivals, as Walter Mazzarri begins his second reign at troubled Napoli.
Napoli continued their struggles on home turf with Sunday's shock 1-0 defeat to Empoli which left the Serie A champions already well off the pace in this season's title race.
Khvicha Kvaratskhelia helped put Napoli's Serie A title defence back on track with a brace in Saturday's 3-1 win at Verona which moved the champions four points behind leaders AC Milan.