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The Greek second seed dropped the second set but won in a tight decider on his third match point, securing a 6-4, 4-6, 7-5 victory.
The 22 year-old, who beat Rafael Nadal en route to the Australian Open semi-finals last month, will face Russia's Khachanov on for a semi-final spot.
"I played really well. Both of us, we played a high level of tennis and didn't give many break-point opportunities," Tsitsipas said. "There were pretty good serves from both sides and I am really glad that I fought so hard towards the very end. It was exhausting, but it paid off."
Tsitsipas, playing his first ATP Tour event since losing to Daniil Medvedev in the semis in Melbourne, clinched the opening set by breaking to love in the 10th game.
He was pushed all the way by the impressive Hurkacz, but despite squandering a match point in game 10 of the third set, he sealed a last-eight spot two games later.
The world number six is the favourite for the title, after the early exits of top seed Medvedev and Alexander Zverev in the Netherlands.
Earlier on Thursday, Zverev's conqueror Alexander Bublik lost 6-7 (5/7), 6-3, 6-1 to American Tommy Paul.
Frenchman Jeremy Chardy ended David Goffin's hopes of back-to-back ATP titles with a 7-6 (7/3), 7-6 (7/5) win.