Raducanu, the British number three, took the opening set after a gruelling 71 minutes, despite sacrificing a 4-0 lead in the tie-break.
She then held a 3-0 advantage in the second set but Kostyuk, who had lost five singles matches in a row since May, rallied to break and draw level at 4-4.
Having forced Kostyuk to serve to stay in the match, the Ukrainian double faulted to hand Raducanu two match points, which the Briton took on her second chance.
The 2021 US Open champion said it was an “extremely good win”.
“I’m happy with the way I fought through the first set and in the second, stayed tough when I needed to,” she added.
Norrie, meanwhile, managed only three winners in the first set, two of them on aces, but dominated the second to force a decider.
After saving five break points in the fifth game of the final set, he broke Musetti, who had a first-round bye, in the sixth to take a 4-2 lead.
The Italian saved two match points in the final game – one on a Norrie double fault – but then sent a backhand beyond the baseline, handing the Briton victory after two hours and seven minutes.
Admitting he was playing “like a kid again”, Norrie said: “I just kind of let go of the expectations and I’m playing tennis more to enjoy it now.
“You’ve got to enjoy it otherwise you can get burned out on this long schedule.”
