De Minaur beat Learner Tien on Thursday (AEDT) to progress to the last eight, and he'll now meet Berrettini for a spot in the semi-finals.Â
World No.48 Berrettini was leading 5-2 on Friday (AEDT) when his sixth-ranked opponent retired, fighting through tears from his chair during a change of ends.
It's already a win for de Minaur, of sorts, as he chases a spot in the prestigious and lucrative ATP season finale in Turin for a third successive year.
There are eight spots available and de Minaur was ranked 10th starting this week.
But with the now-second-ranked Jannik Sinner to miss the ATP Finals with injury, the Australian's chances improved.
They have continued to improve in Austria, with not just his own hard-earned points by making the quarter-finals but with Cobolli, who was sixth in the standings, now out of the tournament.
De Minaur has a great opportunity to continue to rise.
He was also assisted by Spain's Rafael Jodar losing in the last 16 at the Swiss Indoors in Basel.
With a victory over Berrettini, de Minaur can leapfrog Jodar into ninth in the live ATP rankings.
Top seed Alexander Zverev also won through to the quarter-finals in Vienna, beating Tallon Griekspoor 6-4 7-5.
Victory confirms Zverev's ascent to world No.1 when the rankings are updated on Monday, making the reigning French and US Open champion the second German after Boris Becker to reach top spot.
"To have the feeling to be on the top of the world is amazing," Zverev said.
"It is different than winning a tournament or a title, but it's such a special feeling to know that the No.1 is going to be next to your name when the new rankings come out."
Zverev is on de Minaur's side of the draw in Vienna and they will meet in the semi-finals if they both win their next match.
