Kostyuk launched her latest bid for a maiden Grand Slam title by breezing past Hunter 6-4, 6-1 in the first round.
After stumbling at this year's Australian Open, Kostyuk has been in sublime form at the Grand Slams, making the semi-finals of the French Open and Wimbledon to arrive at Flushing Meadows as a genuine contender in the women's draw.
The 11th seed got down to business quickly once the match began and powered to a 4-1 lead before overcoming a wobble to close out the first set when Hunter produced a double fault on set point.
Kostyuk tightened her grip on the clash in the second set, breaking early and never allowing Hunter a route back into the match as she wrapped up victory in 67 minutes.
"I'm very happy to be back to this amazing place," Kostyuk said on court.
"It's not easy to play a qualifier, they've played a whole tournament (already). I'm very happy with my performance today. I try to keep myself grounded. There's still a long way to go, I'm enjoying my tennis."
The 24-year-old faces 2017 champion Sloane Stephens or Clara Tauson next.
Svitolina cruises
Svitolina followed in Kostyuk's footsteps in the evening session with an equally dominant performance, beating Argentine Solana Sierra 6-1, 6-2 in just 51 minutes.
The ninth seed, who reached the U.S. Open semi-finals in 2019, produced an authoritative display to brush aside the world number 91 and begin her campaign on a high.
"It was a great match, it was a pleasure playing at night, it brings the best out of me," Svitolina said on court.
The 31-year-old arrived in New York after a strong North American hard-court swing and has now won 43 matches this season, with her aggressive game complementing the defensive qualities that have long been a hallmark of her tennis.
Svitolina also enjoyed a memorable run in mixed doubles alongside husband Gael Monfils, with the pair reaching the semi-finals before losing to Belinda Bencic and Flavio Cobolli.
"It was a great experience. Of course it was nerve-wracking kind of matches and battles. We had some great wins," she said.
"I was a little bit upset with the loss in the semis. But we really enjoyed this battle. It was our first and last time unfortunately."
For 22-year-old Sierra, the night marked her US Open main-draw debut after qualifying for the tournament for the first time.