Draper, the 25th seed, becomes the first British man to reach the semi-finals of the US Open since Andy Murray in 2012, as well as the first Briton to make it to the last four of a Grand Slam since Cameron Norrie at Wimbledon in 2022.
Having beaten unseeded opponents in his four previous matches at Flushing Meadows without dropping a set, Draper faced his biggest test against the Australian, who won their previous three meetings.
However, he had little trouble against De Minaur, who appeared to be hampered by a hip injury, as he cruised to victory in just over two hours to set up a semi-final meeting against either Jannik Sinner or Daniil Medvedev.
"I think I played a solid match. I feel the best fitness-wise I've been in a long, long time and I think that's where Alex has sort of got me in the past," Draper said in his on-court interview.
"I also think he was struggling a little bit today with something, which may have helped me. Credit to Alex. He's an amazing fighter, an unbelievable player and we're going to have many more battles to come."
De Minaur struggled to find his range from the start and was broken early, setting the tone for the match as he racked up the unforced errors and Draper raced into a 4-1 lead.
Finding himself on the ropes, the Australian ratcheted up his forehand power, but Draper served well to clinch the first set.
With Draper up a break and leading 2-1, the second set was briefly interrupted when both players had injury concerns.
De Minaur, who pulled out of a Wimbledon quarter-final in July due to a hip injury, appeared to touch his hip while moving gingerly, while Draper required a timeout to have his right thigh strapped.
When play resumed, De Minaur broke to level the set at 4-4, but Draper recovered from his injury well and broke in the 11th game before consolidating to seize a 2-0 lead in the match.
De Minaur was unable to shake off his own injury, however, and his fitness continued to deteriorate as the match wore on.
He could not compete with Draper's power and precision in the third set as he was broken twice in succession before losing in a Grand Slam quarter-final for the fourth time in his career.
Draper next faces another tough test in either Australian Open champion Sinner or 2021 US Open winner Medvedev, who meet later on Wednesday.
"I'm a big sports fan, a big tennis fan, so I'll definitely be watching the match tonight if it doesn't go too late," Draper said.
"And yeah, I'll just enjoy the win and relax."
