Britain’s third-ranked Raducanu won the opening after a tough 71 minutes, despite taking a 4-0 lead in the game.
She then led 3-0 in the second set, but Kostyuk lost five games in five straight singles games since May, breaking through and drawing with a 4-4 record.
Ukraine’s double shortcomings forced Kostyuk to stay in the game, taking Raducanu’s two matches and the British seized her second chance.
The 2021 U.S. Open champion said it was “a great victory.”
She added: “I’m happy with the way I fought in the first set and I need to keep it tough in the second set.”
Norrie, meanwhile, had only three winners in the first set, two of whom won the ACE but dominated the second set forcing the decision-maker in the second set.
After saving five breakout points in the fifth game of the final, he broke the first round goodbye Musetti to lead 4-2.
The Italian saved two match scores in the final match – one in Norri’s double fault – but then sent a backhand outside the benchmark, passing the British victory two hours and seven minutes later.
Norry admitted he was playing “like a kid again” and said: “I just gave up on expectations and I was playing tennis more now.
“You have to enjoy it or you can burn down on this long schedule.”
