Palermo's volatile president Maurizio Zamparini said coach Delio Rossi had ruined his team and rated his chances of staying at "one percent" after an extraordinary game that saw the visitors race to a 5-0 half-time lead.
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Inter's win lifted the defending champions into second place in Serie A, at least for 24 hours, one point above Napoli and two behind leaders AC Milan.
Udinese and Roma, held 2-2 by Parma in Vincenzo Montella's first home game in charge, gained ground on fourth-placed Lazio, who lost 1-0 at Cagliari following a comical own goal from Brazilian defender Andre Dias after 40 minutes.
"Rossi has a one percent chance of staying. He has ruined my Palermo," said Zamparini. "Rossi has destroyed this team."
Rossi refused to quit but added: "It is solely my fault. I prepared badly for the game."
European champions Inter bounced back from Wednesday's 1-0 Champions League defeat at home to Bayern Munich by scoring twice in the last 20 minutes against an ailing Sampdoria.
Cameroon striker Eto'o cut inside his marker to rifle home the second in stoppage time.
Roma coach Montella restored Francesco Totti to the starting line-up after controversially putting his captain and ex-striking partner on the bench for Wednesday's 1-0 win at Bologna.
