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Francesco Giudolin's side took the lead 10 minutes before the break through Danilo, who seized on the ball in a packed penalty area and smashed home.
Cagliari came close to an immediate response when Agim Ibraimi turned a free kick inches wide, with the visitors unable to respond before the break.
The Sardinian club then saw hopes of a comeback fade early in the second half, thanks to a special strike from Di Natale.
The 36-year-old curled in a free kick on 54 minutes, leaving Cagliari keeeper Michael Agazzi rooted to the spot as his strike sailed into the top corner.
Cagliari enjoyed plenty of possession late on but were unable to create any clear-cut chances, with the Zebrette holding on to claim all three points.