Kyriacou fired a four-under 68 in the opening round of the Chevron Championship in Houston, Texas, to be one shot shy of Taiwan's Peiyun Chien.
Fellow Australians Minjee Lee and Karis Davidson shot two under.
But compatriots Hannah Green and Grace Kim struggled to three-over rounds while Sarah Kemp finished at four over.
Kyriacou landed three birdies on the front nine and a bogey at the 10th before finishing with a flourish when her wedge approach went in.
"I had 89 metres in, it was a perfect wedge shot for me so me and my caddy were pretty confident," she said.
"He was like 'come on, one more hole, try to stick it in there close'.
"There was a crowd and everything and I was just trying to get it close and then it went in which was exciting for me since that's never happened in a pro event.
"It was a great feeling."
The 22-year-old Sydneysider is among a batch of six golfers including former world No.1 Nelly Korda at four under.
Taiwan's Chien, the world's 189th-ranked player, stole the spotlight from the sport's big names by mixing six birdies with a single bogey at the Jack Nicklaus-designed The Club at Carlton Woods.
With the tournament moving to Carlton Woods after calling the Mission Hills Country Club in Rancho Mirage, California, home for 51 years, the nine past champions in the field, including world No.1 Lydia Ko and last year's winner Jennifer Kupcho had no home course advantage.
Ko had a roller coaster round including four birdies and three bogeys which left the New Zealander at one-under alongside Canada's twice major winner Brook Henderson.
But Chien, playing the back nine first, made the turn at two-under before making her only bogey of the day at the first.
That was the 32-year-old's only stumble as she recovered in style to card four birdies in her final six holes.
"I saw the pin sheet last night ... it feels so hard," said Chien, who came to the major off her best result of the season, a tie for fourth at last week's LOTTE Championship.
"We did very well today."