No British man apart from Murray had made the last eight in Melbourne since John Lloyd in 1985.
Rafael Nadal battled into the Australian Open quarter-finals with a hard-fought victory over a determined Diego Schwartzman.
"That's why I'm in the quarter-finals, because every time I step on the court and I'm playing, I believe I'm going to win. It's certainly my best result at a slam", Edmund said.
Edmund had the worst of conditions on Friday, requiring five sets to beat Nikoloz Basilashvili in 40 degree heat, and while Seppi also needed to go the distance against Ivo Karlovic, it's much easier facing a big server in frequently shorter points in longer matches. It shows I'm improving, also that the hard work is paying off.
He said: "I have to believe it".
The second set gave him scope to recover and, after 45 minutes of tighter control, he was back in the match.
Both players continued to remain assured behind their delivery until the 12th game of the second set when Edmund seized on an indifferent service game from Seppi to break the Italian for the second time and level the match at one-set all.
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"I have to believe I'm going to win and believe in my game". Why, they have asked him, do you waste the first three or four hours of these matches? I feel I have a good ability to control matches and the outcome of matches. You want black and white evidence to say, yeah, I'm playing well and beating these guys.
"I'm just really focused right now on my side of the court". Even when Allertova at game point at 40-0 in the last, Svitolina went on a five-point roll an closed it out with a backhand victor down the line.
It went with serve until Seppi punished Edmund for running round his backhand far too often with a forehand victor and then produced another off the other wing to take it 7-4.
Edmund was clocking up the aces to hold comfortably and Seppi's metronomic groundstrokes were deserting him, but all under major pressure from his opponent, who was cracking the ball deep off both flanks.
The man who lost tight three-setters time and again on the ATP Tour previous year is now finding a way to win when his back is to the wall. The 33-year-old had the first set in lockdown, hardly making any errors and striking the ball with ideal timing.
The Italian's clever changes of direction in the rallies were testing his legs, but still Edmund kept pushing him back and broke for 5-3 when Seppi dumped a backhand in the net. "I played well. He played well".
"To play to win, not to play to avoid losing, to have the heart", said Rosengren. While the ball is in play, he crushes his forehand, he wallops his serve.
Cilic, who defeated Pablo Carreno Busta 6-7 (2/7) 6-3 7-6 (7/0) 7-6 (7/3), reached his second grand slam final at Wimbledon last season and has set himself lofty goals for 2018.