Giacoma Bonaventura, 34, scored his first international goal for the dominant hosts, before Domenico Berardi struck twice.
Davide Frattesi tucked an effort inside the post for the fourth in injury time.
Italy moved three points behind group leaders England, who they face at Wembley on Tuesday night, with three games left.
Defending European champions Italy are level on 10 points with third-placed Ukraine, who beat North Macedonia 2-0, and will need at least two more wins to qualify for Euro 2024.
Victory for England, however, on Tuesday would see the Three Lions win the group.
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