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.
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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.
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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.
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