The 26-year-old midfielder who opened his 2018-19 season goal account in his side's 3-1 win over Schiltigheim a fortnight ago started the rout with a brace in the 11th and 28th minutes to give Antoine Kombouare's men a first-half lead.
Two minutes after the restart, Guinea's Jules Keita extended the lead for the visitors before Sliti completed his hat-trick from the penalty spot in the 50th minute.
Cape Verde international Julio Tavares also got on the scoresheet for Dijon to make it five before Jordan Marie wrapped up the win in the stoppage time.
Wahbi Khazri started from the bench for Saint-Etienne and found it difficult to turn things around after he was introduced for Assane Diousse as his team crashed out of the competition.
Dijon who are placed 17th in the Ligue 1 table now await their next opponent in the round of 16 of the French Cup.