There were doubts about De Minaur’s fitness before the match even started after his practice session lasted just 15 minutes earlier in the day.
Appearing in his fourth Slam quarter-final, the Australian, 25, was sluggish as Draper breezed through the first set and went up an early break in the second.
As Draper clinched that advantage, De Minaur clutched his hip in a potential reoccurrence of the injury that forced him to withdraw from his quarter-final against Djokovic at Wimbledon.
The physio appeared on court at the subsequent change of ends – but that was to treat Draper rather than De Minaur, with the Briton having his upper thigh taped.
Draper did not seem too troubled and soon had five break points for a 5-2 lead, only for each to be saved by his opponent.
That began a run of three consecutive games which threatened to swing the momentum of match in the favour of a resurgent De Minaur.
At that stage it was Draper who was struggling physically but his powerful, swinging left-handed serve returned at the crucial time to halt his opponent’s momentum.
That fearsome strike helped him close out the set and a huge forehand – his other main weapon – earned another break of serve early in the third.
De Minaur mustered one last challenge a game later but after Draper’s serve again saw off two break points at 3-2, the Australian faded badly.
The final set was over in just 32 minutes to make Draper the third British man since Murray to reach a major semi-final.
He will have 48 hours to recover and treat any injury issue before attempting to do what Kyle Edmund at the 2018 Australian Open and Cameron Norrie at Wimbledon in 2022 could not – progress from the last four to the final.