All eyes have been on Draper this week as he takes on the pressure of being the great British hope with Wimbledon approaching at the end of this month.
The 23-year-old has said he is comfortable dealing with that weight of expectation, and underlined that belief with this battling display.
He did, however, start slowly as a run of 11 consecutive points for Popyrin – briefly interrupted by an unexpected sudden gust of wind that sent a hat and bits of paper on to the court – led to the Australian taking the first set.
It was the second match in a row where Draper has taken his time to get into his groove and he revealed he had been struggling a bit with illness this week.
“I’ve not been feeling great actually, to be honest, the last day and a half,” he told BBC Sport.
“I was a bit flat out there. My body feels good but I was a bit under the weather.”
Draper will have his eyes firmly set on claiming a maiden title at Queen’s but he will have another incentive to keep progressing as reaching the semi-finals will mean he is seeded fourth at Wimbledon.
That would mean avoiding facing the likes of defending champion Carlos Alcaraz and world number one Jannik Sinner.
“It’s obviously a big thing,” Draper said. “I think it would definitely help. But at the same time, I don’t think I will think about that at all.
“I can’t control who I’m playing, can’t control any of those things, but I can control what I do between now and the next match, and I’ll prepare the best I can.”