After scraping into the quarters with a 3-6 7-6 (7-5) 7-5 win over French qualifier Quentin Halys, the fifth-seeded de Minaur had his supporters' hearts in their mouths again on Sunday against world No.24 Rublev.
The Australian was comprehensively outplayed in the first set, but produced a stunning comeback to win 1-6 6-2 6-3 in two hours neat.
De Minaur, 27, will now play a semi against world No.39 Hubert Hurkacz of Poland, against whom he has a 3-1 head-to-head record. Hurkacz beat Karen Khachanov 7-6 (7-2) 6-2 earlier in the day.
The 28-year-old Rublev pounced on de Minaur at the outset of their match, breaking him in his first service game while building a 3-0 lead.
After each player held serve Rublev broke de Minaur again to lead 5-1, before holding to 15 for the set.
As comprehensive as that opener was, however, de Minaur flicked a switch to turn the match on its head.
Rublev had to save two break points in a lengthy deuce game to open the second set, but couldn't resist as the Aussie broke him in the fourth game before holding comfortably for 4-1.
A further break, from deuce, in the eighth game gave him the set.
De Minaur was made to fight to open the third set, saving a break point, but broke the Russian straight away to lead 2-0.
Rublev fought back immediately, breaking back to 30 then holding for 2-2, but de Minaur secured the decisive next break in the eighth game to make it 5-3 before serving out to love for the match.
Sunday's other quarter finals will be fought out between top seed Alexander Zverev and sixth seed Novak Djokovic, with third seed Daniil Medvedev playing Argentina's Francisco Cerendolo.
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