"Having one match under my belt was always going to be helpful," Barty said.
"I felt more comfortable out on the court today and to be honest I knew I had to play better to make sure I could match it with Martya.
"She's an amazing player, she truly is, she's playing well beyond her years. It won't be a name that we will forget anytime soon I don't think.
"I haven't played on grass since Wimbledon so it's been a long time. For me, I feel most at home on grass and I was able to click up a gear and really use some of that grass court experience."
Kostyk boasts limited experience on grass and looked anything but comfortable after she fell to the turf in the first set, and Barty began to dominate soon after.
It took just 27 minutes for Barty to claim the first set in comprehensive fashion and she was just as dominant in the second, saying there is no better surface to play on than grass when the sun in shining.
A day after being pushed by world No. 883 Lyudmyla Kichenok, Barty was on another level with a host of deft touches making for a far more convincing start to the day.
Kostya left the court after the opening set with the teenage sensation overwhelmed by a Barty masterclass in a clash punctured by a host of Ukrainian mistakes.
The rising Ukranian star asserted herself early in the second set but ultimately proved no match for the firing Ipswich product as the contest wore on.
Aged 21, Barty has been, and to an extent still is, in the same situation as Kostyuk - a star on the rise carrying a country on her back.
"She's handling it a hell of a lot better than I did to be honest," Barty said.