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Australians John Millman and Alex de Minaur continued their run of good form at the Sofia Open as they booked hard-earned places in the quarter-finals.

The fine end-of-season campaigns being enjoyed by Australia's top two men players, Alex de Minaur and John Millman, continued at the Sofia Open as the pair battled their way into the quarter-finals.

Millman, fresh from his first ATP Tour triumph at the Astana Open at the start of the month, defeated veteran Gilles Simon 7-5 6-7 (3-7) 6-2.

Then De Minaur, the No.1-ranked Australian man, followed up with a 6-4 7-6 (7-5) victory in his second round match with Russian Aslan Karatsev.

The 31-year-old Queenslander Millman again demonstrated his current confident form by prevailing against Simon who, in typically doughty fashion, saved four match points before eventually capitulating after a draining three hours and 18 minutes.

"I feel tired now," admitted Millman, who could have saved himself an hour's extra work if he could have converted two match points at the end of a marathon 82-minute second set.

"I've played with Gilles in Sydney before and hit with him a lot. He is so physical and hard to hit winners against. You know you're going to be in a bit of a war against him," Millman admitted afterwards.

"He's one of the better indoor court players of the last 10 years so this is a really nice win for me to have. I'm happy I won, happy I can stay here a little bit longer."

The sixth-seeded Millman brushed aside the loss of the second set by earning another break at the start of the third.

There was more defiance from Simon, who saved two more match points when 5-1 down in the decider, but at the fifth time of asking Millman sealed the win and a quarter-final date with Canadian Vasek Pospisil.

The pair met last year in the Davis Cup finals with the Canadian coming out on top in straight sets.

"I will definitely need that energy against Vasek, he's another really good indoor hard court player," said Millman.

De Minaur, the No.3 seed who has also been in good nick having reached the final of last month's European Open in Antwerp, always looked to have the easier task against the world No.114 Karatsev but it was far from a straightforward victory.

An immediate break to love smoothed De Minaur's comfortable path to the first set but the Russian qualifier hit back to earn two set points in the second.

Once the 21-year-old Australian had rescued those and taken the set into a see-saw tiebreak, the pair eked out no less than nine mini-breaks between them before De Minaur finally sealed the win in one hour 52 minutes.

Having saved six of eight break points, the Australian was delighted to move through. "He was a very, very tough opponent," De Minaur said of Karatsev. "It was an incredibly tricky match and I'm obviously very happy with the win."

Next up for De Minaur is a fascinating rematch with Italy's rising talent Jannik Sinner, who beat Swiss lucky loser Marc-Andrea Huesler 6-3 6-4.

De Minaur and 19-year-old Sinner met in last year's Next Gen ATP finals when the teenager won convincingly.