The Dane, who is still searching for an elusive Grand Slam title to support her top ranking, on Monday 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 a gruelling 3hr 2min encounter which finished 2330 local time (0330GMT).
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.