Duvan Zapata headed Atalanta in front from an Alejandro Gomez free kick in the 19th minute and Gomez himself added the second with a deflected shot three minutes before halftime.
There were three goals in a seven-minute spell early in the second half as Alfred Duncan pulled one back for Sassuolo, Gianluca Mancini restored Atalanta's two-goal lead and then Duncan struck again.
With the game on a knife-edge, Ilicic was brought on in the 63rd minute and the Slovenian became the first player to score a hat-trick in Serie A from the substitutes' bench since 2011.
He opened his account with an angled shot in the 74th minute, added a second with three minutes left following an exchange of passes with Gomez and completed his hat-trick on the counter-attack in stoppage time.
(Writing by Brian Homewood; Editing by Christian Radnedge)
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