Sixth seed Kvitova was off her game early on, committing 20 unforced errors during the first set - twice as many as her Ukrainian opponent - as she struggled with her serve.
"I was really fighting a little bit [with] the opponent but with myself as well. There was a lot of nerves over there," the 30-year-old Czech said.
"So I was really fighting and, you know, was really tight, and I made some easy mistakes."
The twice Wimbledon champion committed a fifth double fault to hand Kozlova a break point and a 3-2 lead in the first set but recovered to save set point at 5-4 and eventually took the opener on a tiebreak.
She got back on track in the second set to break Kozlova's serve in the first game before firing off consecutive aces and a pair of winners en route to a 2-0 lead.
A relieved Kvitova, who dropped only one point on serve in the second set, wrapped up victory with a powerful serve that a beleaguered Kozlova was unable to return.
She will next face either Kirsten Flipkens of Belgium or American Jessica Pegula in the third round.
Kvitova, who reached the quarterfinals of the Australian Open in January, said that winning the tiebreak and getting the break at the start of the second set helped her relax.
"In the second set I think I really played a good one. I served much better. You know, I was hitting clearly not that many mistakes," she said.
The world number 12 has had little match practice since the sport returned from its lengthy suspension due to the COVID-19 pandemic.
After a first-round bye, she lost in the second round of last month's Western & Southern Open, for which Flushing Meadows served as temporary host this year.
