The 32-year-old Kanepi, a former world number 15 whose career was almost ended by illness and a crippling injury to both feet, will face either Ukrainian fourth seed Elina Svitolina or Madison Keys of the United States for a place in the semi-finals.
It will be her sixth quarter-final at the majors and second in New York after also reaching the last eight in 2010.
"It's very tough to believe where I am now after all that has happened. I didn't expect it," said Kanepi, who is playing in the main draw of an event for the first time this year.
At one stage, she even considered quitting the sport before deciding to give tennis one more go.
"It's very important to do what you love. If you do, then things will come," she said.
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