In steamy conditions, the Czech, seeded 14th this year having dropped out of the top 10 for the first time in three years, escaped a first set tiebreak and then broke twice in the second for the victory on the Arthur Ashe Stadium court.
Kvitova, a quarter-finalist at Flushing Meadows last year, has often struggled with the humidity at the year's last grand slam and admitted it was anyone's guess how far she could go.
"You never know - I never know," the 26-year-old said. "I was sick last week but am getting better every day and I feel good on the court and am feeling the ball well."
World number 66 Buyukakcay, who became the first Turkish woman to win a WTA event in Istanbul earlier this year, achieved another first this week as she became the first woman from her country to appear in the main draw at the U.S. Open.
And the 26-year-old Buyukakcay battled Kvitova all the way, including in the first set when she came back from 5-2 down to force a tiebreak.
"She is a big fighter," Kvitova said of her opponent. "I knew that I had to play aggressive, I was tough in the tiebreak, especially when I was down 2-0."
"She’s making history in Turkey and we’re glad tennis is growing everywhere."
Kvitova now plays the number 22 seed, Elena Svitolina, after the Russian beat American Lauren Davis 6-1 4-6 6-3.
(Editing by Frank Pingue)
