Kim, looking for her second major championship title after winning the Evian Championship in 2019, made a fine start to her round with four birdies before she bogeyed the par-4 ninth.
She managed just one more birdie, on the penultimate hole, but was delighted that she avoided bogeys on the back nine.
"I was quite concerned about the front nine, but I was able to do a good job," she said.
"Then on the back nine I had some risky stretches, but I was able to make par each time, and I'm very satisfied with my overall performance to the end," she added.
Kim was runner-up in the 2018 US Women's Open, losing in a playoff to Thailand's Ariya Jutanugarn at Shoal Creek, Alabama, but said that contest was far from her mind.
"This is a new tournament, a new golf course, so I had a new mindset," she said. "My good performance today, I think that gives me confidence," she said.
Lin is hunting for her first major after picking up seven wins on the China LPGA Tour and two victories on the European Tour.
She made five birdies and a bogey on the ninth, her last hole, but was pleased with the way her short game had ensured she avoided further bogeys.
"I feel like my game wasn't at the best, especially first couple holes I struggled get on the green, even from the fairway.
"But I saved some really, really good pars, so I think that takes some pressure off my shoulder to have a solid start," she said.
Lin missed out on the Women's PGA Championship two weeks ago in Baltusrol, finishing third after having a share of the lead going into the final hole.
She has had three top-three finishes this season as she looks to finally get her first win on the LPGA Tour.
Keep it simple
"I feel like when you're on a good run, you better just get something out of it,” said Lin, who had three runner-up finishes in 2022.
“After the two missed cuts I regrouped myself, set a new goal, set a new game plan. Obviously, it's working, so I just wanted to keep it going and see what happened.
"I was definitely trying too hard a couple months ago, thinking I need extra skill to able to win the tournament.
"Then we kind of calmed down and thought of last year, one of the best seasons I've had, and all I did whole year. There were no swing changes. I kept things simple. Kind of remind myself what brings me here," she said.
Irish pair Leona Maguire and Aine Donegan, the latter an amateur, were among six players a shot back of the lead on 69.
Maguire has had three top-10 finishes in majors since 2021 but the 28-year-old is also looking for her first title.
Two bogeys, on the first and the 11th, cost her the chance to hold the lead but she was pleased with the way she recovered.
"I played really solid all day. Hit a lot of fairways, which was key out there I feel like. Missed a few greens, but made some really nice up-and-downs when I needed to keep the momentum going.
"Always nice to finish with two birdies in the last four," she said.
Women's PGA Championship winner Yin Ruoning of China was three shots off the lead after making a one-under 71 while defending champion Minjee Lee of Australia shot an even-par 72.
