The Kiwi Olympic gold medallist shot a solid 1-under-par on the third day, putting her in a tie for fourth and within striking distance of Korean leader Jiyai Shin.
Shin carded a remarkable 5-under-par 67 to claim a one-stroke lead after World No. 1 and third second-round leader Nelly Korda faltered.
A former World No. 1 herself, and two-time winner in this tournament (2008, 2012), Shin collected seven birdies against two bogeys on Saturday under breezy conditions at the famous Old Course.
"I have a lot of experience with links course, and with not like this tough wind, but pretty much I have a lot of good experience," Shin said.
"Really, the wind is nice to me today, and also, I made ... two bogeys today, but I think everyone can make one or two bogeys. So this is nothing. I just keep focused forward."
Shin, 36, has a three-round total of 7-under 209 and is bidding for her 12th victory on the LPGA Tour.
She sits one shot ahead of defending champion Lilia Vu (71 on Saturday), who notched her fifth birdie on the 18th hole to inch closer to Shin.
All told, Vu totalled five birdies to offset two bogeys and a double-bogey on the 13th hole.
"The wheels were falling off in the middle of it," Vu said. "I was getting anxious, but caddie (Cole Pensanti) was able to give me words of encouragement that I'm playing well. It's just a couple bad breaks."
Korda (75) plummeted into third place at 5-under after carding five bogeys and a double-bogey on the 16th hole. Korda, who salvaged her day with a birdie on the 18th hole, entered Saturday with a three-stroke lead over both Vu and England's Charley Hull.
"It's nice to finish with a birdie, but it wasn't the best of days," Korda said. "But it's okay."
Bidding for her seventh title this season, Korda rests one stroke ahead of South Korean Jenny Shin (70) Ko.
"I think it's going to be a tough day (Sunday)," Korda said. "I think the winds are going to be high. There could be rain, as well. I'm going to keep a positive attitude. Take it one shot at a time. I played really well the first two days, so I'm going to take that momentum into tomorrow."
China's Ruoning Yin (73) carded four bogeys against three birdies to drop into a four-golfer logjam at 3-under.
Hull (75) recorded a double-bogey on the par-5 fifth hole during a disastrous round. She sits tied for 10th place with top-ranked amateur Lottie Woad of England and three others.
Woad, who recorded her second round of 72 for the tournament, attends Florida State and is making her Women's Open debut and playing in her fourth major.
