Green is tied for fourth after Saturday's efforts (Sunday AEST) and produced a sparkling front nine which included three birdies and then an eagle on the par-5 eighth.
The Australian handed a shot back on the back nine with a bogey on 11, a birdie on 16 and then another bogey on the 18th when the rain was at its heaviest for the day.
"I'm going to have to do exactly what I did on the front nine — put myself in good positions off the tee as well as on the greens," Green said.
"I felt like on the back nine I had a couple of opportunities to make a birdie but it was a downhill putt, so you just can't be aggressive."
Thailand's Jeeno Thitikul is the player everyone's chasing in New Jersey and she made nothing worse than par over her last 16 holes for a 2-under 70 and a two-shot lead.
Thitikul started with a three-shot lead and it was gone quickly after a bogey on the second hole and the fast start by Green.
But Thitikul kept her cool and kept marching along on a Mountain Ridge course softened by spells of rain, but still difficult enough with its contoured greens to present problems.
Thitikul got up-and-down for par with a nifty pitch on the 18th to finish at 10-under 216, two shots over Celine Boutier of France, who had a 67.
"It's a new day and I don't know what's going to happen, but just going to let golf be golf and then let myself be committed to golf and then just go by the flow with it and then just give myself a lot of chances," Thitikul said.
"Another opportunity under my belt that I have in the final round again."
Hye-Jin Choi had a 66 and was three shots behind.
Green was only the first of the contenders who got within one shot of Thitikul with a four-hole stretch on the front nine that she played in 4 under, including the eagle on the par-5 eighth. But no one could surge ahead.
Ruoning Yin of China was within one shot playing the par-4 15th when she got in trouble off the tee and had a full short iron into for her fourth shot. She hit that to 5 feet and looked as though she might escape with a bogey. But she shockingly three-putted for triple bogey.
Yin answered with a birdie on the par-3 16th with its accessible front pin and had to settle for a 68, leaving her in a tie for fourth at at 6-under 210 along with Green and Allisen Corpuz, who has not won since the US Women's Open at Pebble Beach in 2023. Corpuz shot 68.
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