Trailing by a set and a break to the veteran Italian, the 23-year-old made up his mind to attack and it paid handsome dividends as he recovered to win 6-7 (4), 7-5, 6-2, 6-3 and set up a quarterfinal against either Nick Kyrgios or Grigor Dimitrov.
Edmund, the lone Briton in the men's draw after the withdrawal of Andy Murray, slammed 25 aces on Hisense Arena. His win means he is projected to rise from his current mark of 49 to a career-high ranking of 35. And he may not be done yet.
"It's a good feeling," Edmund said. "I didn't feel I got off to the best start. In the second set I tried to shift the momentum. I think when I broke in the last game of the second set, I took control of the match. Through to my first quarterfinal, I'm very happy."
Edmund said he didn't feel any extra pressure because of the absence of Murray.
"I'm just very happy, keep doing my best," he said. "I know people at home are waking up silly hours in the morning, so I'm very proud for that."
It was a resilient, impressive performance from Edmund, who had beaten 11th seed Kevin Anderson in round one. Both he and Seppi had been forced to the limit in their previous matches, going the distance in brutal temperatures that reached 40-degrees Celsius (104 Fahrenheit), but the Briton was the stronger as the match wore on and he took his chances.
In the early stages it was Edmund who looked to showing signs of fatigue after his third-round win over Nikoloz Basilashvili of Georgia, choosing not to run for a number of balls and requiring treatment on his right shoulder midway through the first set.
While Seppi, who was also trying to reach a Grand Slam quarterfinal for the first time, was holding serve with ease, the 23-year-old Edmund saved five break points in the eighth game, coming up with some big serves when he needed them. Two games later, he held from 0-30 and and he won another tight game to force a tiebreak. But Seppi, who had been the better player throughout, won it 7-4 to move ahead.
When Seppi broke in the third game of the second set, it seemed like Edmund might just drift away. But that was when the Briton told himself to attack, and a series of huge forehands helped him to break straight back for 2-2. Suddenly it was Edmund who was looking the fresher and at 6-5, Seppi sent a forehand wide, off the net, as Edmund levelled the match.
The third set was all Edmund as Seppi began to tire. Stepping in to attack the Italian's second serve, an inside out backhand winner helped him break for 2-0 and a second break in the eighth game gave him the set.
Seppi saved three break points at 1-1 in the fourth set and then had treatment on his right shoulder after the following game. Two more break points went begging for Edmund at 3-2 but he finally broke through for 5-3 when Seppi netted a backhand. Edmund then served out the match with ease, clinching victory with another unreturnable serve.