Nakashima sent over 11 aces and offered up a subdued celebration for the career-breakthrough win in Montreal. He will next play the winner of the match between Spaniard Rafael Jodar and Frenchman Arthur Fils.
Nakashima, seeded 28, converted a break point in the fourth game before breaking his opponent to love to close out the first set, as the visibly frustrated Darderi's composure unravelled.
Darderi dug in his heels in the second set but could only hold off the determined Nakashima for so long, as the American broke him on the fourth try in the penultimate game.
Nakashima sent 19th-seed Darderi skittering around the court in the final game, forcing him into an error at the net on match point and pumping his fists in celebration after the customary post-match handshake.
The men's quarter-finals began a day late after inclement weather forced the postponement of the Monday programme.
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