Fifth seed Rublev had to do it the hard way, having been 5-2 down in the final set at Rod Laver Arena and needing to save two match points to take contest into a 10-point tiebreak of unrelenting tension.
He slipped 5-0 behind in the breaker but roared back to lead 9-7 before ninth seed Rune saved two match points in a late twist.
It took the net cord to separate them, with Rublev's final return clipping the net and popping over on the third match point, leaving the Dane no chance.
"(Tennis is) not a rollercoaster, it's like they put a gun to your head," Rublev said on court.
"I think a rollercoaster is a lot easier, man.
"I never in my life was able to win matches like this, this was the first time in my life I was able to win a match like this.''
Rublev will meet the winner of nine-times champion Novak Djokovic and home favourite Alex de Minaur, who play in the night session.
"No-one wants to face Novak, I'd rather be in any other part of the draw," added Rublev wryly.
Rising star Rune was a point away from becoming the first Danish man to reach the Australian Open quarter-finals and bowed out wondering what might have been.
While not matching his sparkling quarter-final run at last year's French Open, he leaves Melbourne with reputation enhanced.
He had the seasoned Rublev, now a seven-times Grand Slam quarter-finalist, on the ropes.
"The luck was on his side," Rune told reporters.
"It was a great match. We both fought until the end.
"Of course, he went for it. I was maybe a bit passive on a few points, but he could have missed a few."
NO CLASSIC
It was by no means a classic encounter, with the finish adding gloss to a contest in which both players blew hot and cold.
There were five breaks of serve each, and Rune racked up 12 double-faults.
After winning the second set to level the match, Rune was unable to run away with it.
He whacked the net with his racket in frustration as his serve malfunctioned and he imploded in the third set.
Losing 11 points in succession, Rune double-faulted to gift Rublev three break points and thumped a forehand long to fall 4-2 behind.
Rublev served out the set to love, sending Rune off court to regroup.
The Dane was in bother early in the fourth set and had his vital signs measured during a medical time-out.
He took electrolytes, though, and his hang-dog demeanour disappeared as he broke Rublev.
Rune rolled on to a 5-2 lead and levelled the match again when Rublev pounded a drive-volley well long on set point.
The Dane kept the momentum into the fifth, breaking Rublev in the fourth game after stretching for a splendid backhand volley down the line.
"You're in his head, Rune!", a man yelled from the terraces.
It seemed a fair assessment at the time but proved premature as Rune was broken to love when serving for the match.
Rublev hung tough, saved two match points when serving to stay in the match at 6-5 and showed huge grit to stay alive in the tiebreak.
"During the tiebreak, I was thinking for sure it's over," he said.
"(The net cord) was probably the luckiest moment of my life."
