The Johannesburg-born Edmund is the first British man other than Murray to reach the quarterfinals here since John Lloyd was beaten in the quarters in 1985.
The Danish seed is set to take on Carla Suarez Navarro who reached the quarter-finals of the Australian Open for the third time.
"It's a good feeling", Edmund said.
While Seppi, who was also trying to reach a Grand Slam quarterfinal for the first time, was holding serve easily, the 23-year-old Edmund saved five break points in the eighth game, coming up with some big serves when he needed them.
"Definitely pleased. Shows I'm improving".
"In the second set I tried to shift the momentum and once I broke him in the last game of the second set I took control of the match". The world No 3, who recorded the biggest victory of his career when he won the title at the year-end ATP Finals in London in November, beat Edmund in three sets in Washington last summer and was again taken the distance when he beat the world No 49 in Brisbane earlier this month.
Edmund spent his off-season at his new base in the Bahamas working on his serve in particular, and 25 aces played a huge part in the win over Seppi.
"To play to win, not to play to avoid losing, to have the heart", said Rosengren. "I really think he's going to have a great year". Two weeks ago I lost against him.
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Edmund broke him in the eighth game, aced him for the 26th time as her served for the match and, playing big shots all the way to the end, finished it with another big serve wide to the forehand side.
"I guess I get maybe more messages on social media and stuff", he said.
Even without factoring in fitness assessments, the numbers stack up for backing Seppi at prices in any case - there really isn't too much to split the pair from a hold/break perspective - so I'm more than happy to recommend the Italian as a heavy underdog here.
"It was a close first set and I didn't feel I got the best start". You have to play seven matches, five sets, if you want to win this tournament. If there is a chance to stretch a match to five sets, Seppi will jump at it.
"I thought he actually came out very well and timed the ball well, dictating probably more points than I was". Rybarikova double-faulted on break point to open the second set and start Wozniacki on her way to a dominant victory in 63 minutes.
"It was really good to turn it around".
When Seppi broke in the third game of the second set, it seemed like Edmund might just drift away.
Since almost being eliminated in the second round by Jana Fett - Wozniacki trailed 5-1 in the third set before winning the last six games - she has won her next two matches in straight sets.