The British number two, ranked 49th, recovered from a slow start to win 6-7 (4-7) 7-5 6-2 6-3 at Melbourne Park.
"Oh, he's good", Nadal 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.
'It's a shame Andy's (Murray) not here.
Trump shows immaturity by making fun of Durbin
Do I care that he said it at all? "I've always believed that America is an idea, not defined by its people but by its ideals". David Perdue of Georgia - defended the president on Sunday, saying they couldn't recall if Trump made such comments.
His place in the last eight means he is guaranteed a career-best world ranking of around 35.
"Hopefully we'll have more British here in the future but for me just personally I'm very happy to get through".
"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". 'He's a good player, of course, and for me not a surprise. Two games later, he held from 0-30 and and he won another tight game to force a tiebreak. Stepping in to attack the Italian's second serve, an inside out backhand victor helped him break for 2-0 and a second break in the eighth game gave him the set.
Instead it galvanised the Briton, who broke straight back with a forehand victor and began to assume control.
Alex Iwobi on target in Arsenal's demolition of Crystal Palace
And Sanchez's soon to be former team-mates didn't seem fazed by his impending exit either as they raced into a sixth-minute lead. This time it was young winger Iwobi who provided the finishing touch with a clinical close-range finish from Monreal's cross.
If his first-round defeat of 11th seed Kevin Anderson was his biggest scalp, Edmund's sensational performance to beat Seppi was his biggest statement of intent yet.
With a landmark victory in sight, Edmund simply grew stronger, forcing six break points before Seppi finally succumbed with a backhand into the net on the seventh.
But the British No 2 insists flying the flag has not added any extra pressure to his campaign, which has seen him win two five-set matches in sweltering conditions.
After fighting back from a set and a break down having lost the opener in a tie-break, Edmund's massive forehand was in full effect and he was serving impeccably too, finishing with 25 aces.
Holtmann makes Big Ten history as Ohio St. tops Northwestern
Get insight and reaction to Ohio State football and basketball on the Inside the Shoe podcast, brought to you by Land of 10 . To put that in some other perspective, when Meanwell did it for the Badgers, the Big Ten was actually called the Big Nine.