The Dane, who is still searching for an elusive Grand Slam title to support her top ranking, yesterday 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.
"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."
Williams reached the last eight with a 6-3, 6-4 breeze past former world number one Ana Ivanovic and will now face Russian 17th seed Pavlyuchenkova, the youngest player left in the draw.
Williams, seeded a lowly 28 after a year spent battling injury and serious illness, has yet to drop a set at Flushing Meadows this year and Ivanovic never looked likely to break that streak.
"I feel so blessed to be back. A couple of months ago I never expected to be here," said Williams, 29, who came into the US Open with titles in Stanford and Toronto, her first wins after spending 11 months on the sidelines.
Williams said she was looking forward to building on her career tournament victories of 39, which are second only to sister Venus (43) and Kim Clijsters (41) amongst active players.
But she was worried over where she will put any more trophies.
"I don't have any more space. I have a new house in L.A. I created a karaoke room, so I can't put trophies in there," said Williams.
"The other area, the gentleman's lounge, is kinda packed with vintage things I got at a flea market, which is really cool. In Florida, forget it. Venus and I, there's just no more space.
"I would love to keep winning them. But we're like, 'Oh, what are we going to do with this one?'"
Pavlyuchenkova battled back to defeat seventh-seeded Italian Francesca Schiavone 5-7, 6-3, 6-4 to reach her first US Open quarter-final in a mistake-plagued clash.
Pavlyuchenkova triumphed in a match-up of the youngest and oldest players -- 20 and 31 -- left in the women's event.
But the tie, which opened the programme on Labor Day, a national holiday in the US, was not for the purists with 21 double faults, 16 breaks of serve and a combined 95 unforced errors.
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 today 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.