The British number one, playing his first ATP Tour tournament since his run to the semi-finals of the US Open earlier this month, beat the world number eight 6-4 6-4 in Tokyo to make it through to the quarter-finals.
It was his third win against a top 10 player this year, having earlier beaten Carlos Alcaraz and Alex de Minaur.
Next up is a winnable last-eight tie against France's Ugo Humbert, who defeated Brandon Nakashima 6-3 6-2 and Draper will have eyes on a second ATP Tour title of the year.
The 22-year-old was strong from the off, forcing break point in the opening game, only for Hurkacz to serve himself out of trouble.
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