The 36-year-old world No 18, from Harrogate, saw off world No 4 Karim Abdel Gawad 11-8, 4-11, 11-9, 12-10 to book a last eight showdown with Germany’s Simon Rosner.
The Yorkshireman, a former world No 1 and runner-up at the event in 2010, clinched a dramatic victory after 71 minutes having twice squandered match ball.
Willstrop, who plays for Pontefract Squash Club, now has a rest day before facing world No 6 Rosner tomorrow and admitted he needed great mental strength to get past an opponent who had only recently won a PSA Platinum title over world No 1 Ali Farag.
“I am absolutely thrilled,” said Willstrop, who had recently taken time out to perform in a theatre show called Hangmen. “I am on a rush of adrenaline, particularly in these conditions where it is shoot-out squash.
“The intensity is cranked up and you haven’t got any time to settle. The mental intensity of the squash that has been played is heavy, so the adrenaline is high, I didn’t know whether I was capable of doing it.
“I know what I am doing in training, and I know I can play alright and I know I can hit the ball alright, but it is whether physically I can cope with these players in a match situation.
“To be here (the quarter finals), and with a rest day, that is quite positive. It is a fabulous prospect, and hopefully I can get recovered well and do it justice.”