In Inter’s first match since Samuel Eto’o transferred to Russian club Anzhi Makhachkala, the biggest problem for Gasperini was his porous three-man defense.
Palermo captain Fabrizio Miccoli scored two key goals, including one in the 86th minute that broke a 2-2 deadlock, and South Americans Abel Hernandez and Mauricio Pinilla also found the target for the Sicilian squad.
Diego Milito scored twice for Inter and newly signed Diego Forlan pulled one back for the visitors in second-half added time.
New Juventus coach Antonio Conte, meanwhile, had little to worry about as newly signed midfielder Andrea Pirlo set up goals for Stephan Lichtsteiner and Claudio Marchisio.
Simone Pepe and newly signed Chile international Arturo Vidal also found the target for Juventus as the Turin club played its first official match in its new stadium.
Parma pulled one back with a penalty kick from Sebastian Giovinco in added time after Juventus defender Paolo De Ceglie was shown a red card.
Elsewhere, it was: Catania 0, Siena 0; Chievo Verona 2, Novara 2; Fiorentina 2, Bologna 0; Genoa 2, Atalanta 2; Lecce 0, Udinese 2; and Roma 1, Cagliari 2.
Defending champion AC Milan drew 2-2 with Lazio on Friday and Napoli beat 10-man Cesena 3-1 Saturday.
On a warm night in the Sicilian capital, Inter started with 11 foreigners and all three substitutes were also non-Italian.
Milito, who missed much of last season injured, put Inter ahead by redirecting a shot from Dejan Stankovic with a slight deflection in the 30th minute.
Miccoli equalized in the 48th from the center of the area and Milito then restored Inter’s lead with a penalty in the 51st, but Palermo equalized again in the 54th when Miccoli set up Hernandez.
Miccoli curled a free kick over Inter’s wall for the winner in the 86th and Pinilla made it 4-2 by surprising Inter goalkeeper Julio Cesar in the 88th.
On a warm night in the Sicilian capital, Gasperini became unsatisfied with newly signed striker Mauro Zarate and replaced the Argentine with standout playmaker Wesley Sneijder—a curious bench choice—in the 34th.
While Inter’s troubles weren’t all that surprising, Palermo’s success was startling, considering that the club sold standout forward Javier Pastore and goalkeeper Salvatore Sirigu to Paris Saint-Germain over the offseason.
Also, Palermo owner Maurizio Zamparini fired coach Stefano Pioli on Aug. 31, handing the team over to youth squad coach Devis Mangia.
In Turin, Pirlo set up a goal for Lichtsteiner in the 16th and Pepe doubled the lead in the 58th, converting a pass from Juventus captain Alessandro Del Piero.
Vidal scored with a long shot in the 73rd and Marchisio made it 4-0 in the 83rd after another setup from Pirlo.
Juventus played a friendly with Notts County for the ceremonial opening of the new 40,000-seat stadium on Thursday but the Parma game was its first official match. The club has not named the stadium yet.
Juventus forward Alessandro Matri had a goal disallowed for a questionable offside call in the 39th, then hit the post in the 55th as the Turin squad impressed in former Juventus player Antonio Conte’s debut as manager.
Having missed out on Europe with a seventh place finish last season, Juventus is desperate to make the top three this campaign and qualify for the Champions League.
“It was a good debut,” Conte said. “I’m happy with the players and I want to thank the fans, they were like a 12th man. That said, I don’t want to disillusion anyone—we weren’t perfect and there’s still a lot of work to do.”