Palermo had become a top-flight casualty when they were beaten 3-0 by second-tier side Modena on Saturday, but there were no further shocks 24 hours later.
Lazio thumped Bassano Virtus 7-0, while long-serving striker Di Natale scored all five goals as Udinese cruised past Ternana 5-1.
Stefano Poli's Lazio side were frustrated by their lower-league opponents in the first half, as they were unable to break the visitors down.
The second period was a different story, however, as Lazio's class eventually told, with Antonio Candreva opening the scoring a minute after the break.
A brace from Balde Diao Keita followed, either side of a goal from Stefan de Vrij - the defender's first since signing from Feyenoord.
Di Natale clinched his hat-trick in the first period in Udine, before adding another two goals in the final 15 minutes as the hosts comfortably got the better of their second-tier opposition.
Cesena's victory over Casertana was in stark contrast to that of their Serie A rivals, as Pierpaolo Bisoli's men squeezed past their Lega Pro-opponents 1-0, Guido Marilungo's goal allowing all around Stadio Dino Manuzzi to breathe a sigh of relief.
Genoa were involved in a similarly tight affair, as Mauricio Pinilla's 31st-minute penalty saw them advance at the expense of Serie B side Virtus Lanciano.
Elsewhere, Verona, Empoli and Sampdoria all enjoyed comfortable victories.
Verona saw off Cremonese 3-0 thanks to three goals in 17 second-half minutes, while Empoli triumphed over L'Aquila by the same scoreline courtesy of Francesco Tavano's hat-trick.
Sampdoria's Eder matched the feat of Tavano in a 4-1 home win over Como, while elsewhere, Serie B outfit Brescia advanced to the fourth round as Andrea Caracciolo's 88th-minute strike earned a 1-0 victory over Latina.
