After becoming the first British man to reach the US Open final four since Andy Murray in 2012, Draper says he wants to continue to emulate the Scot.
“I see it as a responsibility to be the person who’s performing,” said Englishman Draper.
“Andy has had a lot of success over a long period of time, and my biggest respect for him was that he did that with a lot of pressure on his shoulders, especially around Wimbledon time.
“I see it as a responsibility to achieve great things for myself and for British tennis.”
He will next play the winner of the match between France’s Ugo Humbert and America’ s Brandon Nakashima.
Hurkacz’s exit means five of the eight seeded players have been knocked out, with Taylor Fritz, Casper Ruud, Stefanos Tsitsipas and Frances Tiafoe losing in the first round.
Elsewhere, Katie Boulter progressed to the second round of the China Open with a straight-set win against Taylor Townsend.
The British number one made a slow start in Beijing, failing to take three break points in the opening game and losing her first service game.
Boulter, playing for the first time since a second round exit at the US Open, recovered to win 6-4 6-4.
She will face the winner between America’s Coco Gauff and France’s Clara Burel in the next round.