Draper won the first set against Shelton with a serving performance every bit as commanding as the one he produced in his fourth-round win over Taylor Fritz.
He conceded just three points on serve, and one break was enough to secure the first set.
However, the partisan crowd did not take too long to find its voice, with Shelton’s huge ground strokes helping him win the first three games of the second set.
A frustrated Draper was given a warning for unsportsmanlike conduct after hitting a ball high into the stands in the third game, but he refocused to force the set back on serve.
The match turned decisively in his favour when Shelton double faulted to give Draper a chance to serve for a semi-final spot.
He went 0-30 down but reeled off the next four points to secure victory.
“I felt like maybe the conditions suited me a little bit more,” Draper told BBC Sport.
“His serve would be bouncing very high if it were lively, but I felt it was a challenge for both of us dealing with the wind and the dead conditions.”
However, the Briton expects no such advantage against Alcaraz.
“I think these courts bring out the best in him,” Draper added.
“He can show all his skills and look for his forehand.
“I think the conditions really suit him – but then again, he does well in most circumstances.”
