Draper received treatment from the physio during the second set while De Minaur also appeared to be hampered by an injury.
The Briton said he called for the trainer after he “felt something” in his leg on set point in the first set, but he played down the issue.
“I was little bit worried at the time but I played two sets on it,” he said.
“I did notice his attitude was maybe a little bit subdued, he was carrying an injury of some sort, but I still felt like it was a tough match.”
Despite appearing in his first major quarter-final and playing on the 24,000-seater Arthur Ashe Stadium for the first time, he said he did not feel nervous when serving out the match.
His celebrations were also understated, with Draper carefully rearranging his belongings rather than being overcome with emotion.
“Honestly there weren’t any [nerves],” Draper told Sky Sports.
“Once I got over the second-set hurdle I felt confident.”
Draper said his previous experience playing on Centre Court at Wimbledon – coupled with walks around the cavernous Ashe Arena early in the mornings before the crowds arrive – helped him deal with the occasion.
“Every morning it has been a ritual of mine because I practise so early,” Draper said.
“I have gone and taken a look around while it has been empty.
“I have been going out not knowing I would play but taking a look because it is one of the most iconic stages in the sport.
“I think that helped me get used to it today.”
Draper is now ranked 25th in the world and became British number one in June, before going on to beat French Open and Wimbledon champion Carlos Alcaraz at Queen’s in another key moment in his career.
Against De Minaur he remained remarkably composed during and after the match given the magnitude of the occasion.
“He’s maturing all the time, he understands he can’t go out and burn all his energy and emotions early on,” his coach James Trotman added.
“It’s just a sign that he’s becoming more comfortable on this stage and playing with the best players in the world.”