Edmund and his Italian counterpart were both striving for a career-first quarter-final spot in this tournament, and it was the Brit who earned his place with a 6-7(4) 7-5 6-2 6-3 victory.
Both players sparred in the opening set, and individual errors from both camps produced a topsy-turvy 5-5 scoreline. Edmund made it 6-5, but in doing so forced a medical timeout, requiring treatment on his shoulder.
Seppi dominated the tie-break, biding his time before producing two backhand winners to take the first set, during which Edmund had made 23 errors to Seppi's five.
World number 49 Edmund needed four chances at break point before making it 2-2 in the second set. The Brit then rallied, winning 7-5.
At this point, Seppi was rattled, and Edmund produced an unstoppable third-set performance to take the lead. The British number two won the opening three games, before battling back at 5-2 and then taking the set 6-2.
After clinching 13 winners and all of his break points in the previous set, Edmund had victory in his sights and he took the final set 6-3 to become the only British competitor other than Andy Murray to reach the quarter finals down under since John Lloyd in 1985.