By SABC Sport
18th August 2024
The British number one had come from a set down in his previous two matches, but was unable to repeat the feat against the 15th seed, going down 4-6 2-6.
A tight first set was decided on the only break point as Rune took his chance.
The Dane broke again to start the second set and saved three break points on his serve, seizing his only other opportunity on Draper's serve in the 11th game before serving out for victory.
Rune will play Frances Tiafoe in the last four after Hubert Hurkacz retired due to a calf injury a set down in the quarter-final.
Top seed Jannick Sinner, who avenged last week's defeat to Andrey Rublev in Montreal 4-6 7-5 6-4, will meet Alexander Zverev after the German beat Ben Shelton 3-6 7-6 7-5.
In the women's tournament, Iga Swiatek beat Mirra Andreeva 4-6 6-3 7-5 to set up a semi-final meeting with Aryna Sabalenka, a 6-3 6-3 winner over Liudmila Samsonova.
Home hope Jessica Pegula saw off the challenge of Leylah Fernandez 7-5 6-7 7-6 and meets Spain's Paula Badosa, who eased to a 6-3 6-2 win against Anastasia Pavlyuchenkova.