The Dane, who is still searching for an elusive major title to support her top ranking, saw off the tiring 2004 champion Kuznetsova 6-7 (6-8), 7-5, 6-1 and will now tackle Germany's Andrea Petkovic for a place in the semi-finals.
In the other quarter-final in the top half of the draw, three-time winner Serena Williams will clash with Russia's Anastasia Pavlyuchenkova.
It was a huge struggle for the 21-year-old Wozniacki on a night of fluctuating fortunes on Arthur Ashe Stadium in an encounter lasting a gruelling three hours and two minutes.
She surrendered a 5-2 lead in the first set tiebreak and then slipped 1-4 down in the second set before 15th-seeded Kuznetsova's game fell to pieces.
Wozniacki evened the contest on a third set point in the second set and then clinched breaks in the first and fifth games of the decider.
Kuznetsova, who was undone by 78 unforced errors, saved four match points in the seventh game but Wozniacki took victory with a smart backhand volley.
"I thought about the match we played against each other here in 2009. I was a set and down in the second in that one and still won," said Wozniacki.
"I knew I could come back so I just stepped into the lines and went for my shots, tried to make fewer errors. I am in good shape. I can play for five hours if I have to. I just wanted to keep the rallies going."
Petkovic defeated unseeded Spaniard Carla Suarez Navarro 6-1, 6-4 to reach her first US Open quarter-final, having made the last eight at the Australian Open and at Roland Garros this year.
The first two quarter-finals take place on Wednesday (AEST) with Vera Zvonareva, the second-seeded Russian and runner-up last year, facing Australian Samantha Stosur, the ninth seed.
Italian 26th seed Flavia Pennetta tackles unseeded German Angelique Kerber
caroline wozniacki
caroline wozniacki
caroline wozniackicaroline wozniacki
caroline wozniacki
caroline wozniackicaroline wozniacki
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