The 24-year-old American shared a two-stroke overnight lead with South Korea's Chella Choi and was clear of the field after four successive birdies on the back nine in Illinois.
But a bogey on 17 saw her head down the 18th fairway tied for the lead with defending champion Brooke Henderson.
Kang then sank a birdie to win by one.
"I just trusted in my game and didn't worry about anything else. It's all about being confident and believing in yourself," she said.
Kang, a two-time US Women's Amateur champion, said she had received messages of support ahead of the final day from former Olympic decathlon champion and now reality TV celebrity Caitlyn Jenner, and ice hockey great Wayne Gretzky.
England's highest finisher Jodi Ewart Shadoff, who was just one shot off the lead after two rounds, finished on level par after carding successive rounds of 74 to finish 12 shots behind the winner.
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Danielle Kang won her first major title by clinching the Women's PGA Championship at Olympia Fields with a birdie on the final hole.