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.
Milan host Napoli on Monday.
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.
The scenes in Sicily were remarkable, with Chile forward Sanchez, tracked by Inter, scoring a first-half hat-trick and Di Natale completing the rout with a penalty just after the hour.
Sanchez, applauded by Palermo fans when later substituted, had grabbed his fourth after 48 minutes as Palermo fell apart, with Armin Bacinovic and Matteo Darmian sent off for a second booking and a professional foul either side of the break.
Udinese have been in superb form after a shaky start while Palermo have started sliding down the table.
"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."
Udinese are fifth, a point behind Lazio in the Champions League preliminary round spot.
SLACK SAMPDORIA
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.
Dutchman Sneijder put them ahead in the 73rd minute, curling in a 30-metre free-kick as the slack hosts were still organising their defensive wall.
Cameroon striker Eto'o cut inside his marker to rifle home the second in stoppage time.
Inter goalkeeper Julio Cesar, whose blunder led to the Bayern winner on Wednesday, made two important saves with the score goalless while Sampdoria midfielder Andrea Poli hit the Inter post just before half-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.
Totti, making his 600th club appearance, netted a deba
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