Twenty-year-old Draper had been enjoying another memorable run on the ATP Tour, defeating Fabio Fognini and Dominic Thiem to reach the last eight.
But there his campaign came to an end as Huesler clinched a 6-4 6-4 victory in an hour and 20 minutes. Draper will now head to New York, where he will make his US Open debut against Finn Emil Ruusuvuori.
In another quarter-final, Laslo Djere saved two match points to defeat Richard Gasquet 6-4, 3-6, 7-6(6), booking his place in the final four against Huesler.
Elsewhere, Heather Watson eased to victory over Vitalia Diatchenko to make it through to the final round of qualifying at the US Open. The former British number one, who lost just one game in her first-round match on Wednesday, continued her good form by seeing off 19th seed Diatchenko 6-2 6-3. Watson, a former US Open junior champion, has never won a main-draw match at Flushing Meadows but did make it through qualifying four years ago. Paul Jubb joined Watson in the final round of qualifying with a 4-6 6-1 7-5 victory over Portugal's Pedro Sousa. The 22-year-old came from behind for the second successive match and will take on Dutchman Gijs Brouwer bidding to reach the main draw of an overseas grand slam for the first time. There were second-round defeats, though, for Katie Boulter and Liam Broady. Boulter fell 6-3 6-4 to American Ashlyn Krueger while Broady was beaten 4-6 6-1 6-4 by Dimitar Kuzmanov of Bulgaria.
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