Rublev navigates five-set epic to make Australian Open quarters
In a battle of two former junior world number ones, the Russian broke when Rune was serving for the match, then saved two match points before winning 6-3, 3-6, 6-3, 4-6, 7-6 (11/9) in sapping heat on Rod Laver Arena.
He will face either nine-time champion Novak Djokovic, who is struggling with a hamstring injury, or Australian hope Alex De Minaur for a place in the semi-finals.
The victory put the 25-year-old into the Melbourne quarter-finals for a second time and for a seventh time at Grand Slams.
Despite being world number six Rublev, who has won 13 ATP titles, has never been further.
"It's not like a rollercoaster, it's like they put a gun to your head. A rollercoaster is easier," said the exhausted Russian after the three-hour and 37-minute epic.
"I think I was never able to win matches like this and this was the first time ever that I won something like this to be in the quarter-final.
"I was 5-2 down in the fifth, at 6-5 two match points and then 7-3 (behind) on a super tiebreak. I have no words, I'm shaking."
With both players less-than-clinical on serve, Rublev broke the Dane with a forehand winner to take a 4-2 grip in the first set.
But Rune's returns helped create chances and he broke straight back when Rublev played into the net from the baseline.
A Rune double-fault immediately handed the advantage back to his opponent and this time Rublev made no mistakes in serving out the set in 37 minutes.
Undaunted, Rune clung on in a see-sawing 12-minute opening game in set two and turned the tables to break Rublev for a 3-1 lead, which he never relinquished.
With the sun blazing down, Rublev pounced in the sixth game of the next set, opening up three break point opportunities and he made the most of them.
Down two sets to one, Rune called a trainer for what appeared to be a blood pressure test, but unperturbed got back to work and broke for 4-2 and took it to a fifth.
In a battle of attrition, Rune ground down Rublev to move to 5-2 in the decider, but he failed to serve out for the match.
Rublev then saved two match points to take it to the 10-point tiebreaker, where he bounced back from 7-3 down for a gutsy win.
