Karen Stupples finished three shots off the lead after a round of 63. Laura Davies and Seol-An Jeon shared third place at 5-under-par 65.
Lee, who bested her previous career low on the LPGA Tour by four shots, jumped into the red on the Dell Urich Golf Course at Randolph Park with a chip-in eagle at the par-5 third.
She added a birdie at the very next hole and ran off back-to-back birdies from the sixth to move to 5 under.
At the par-3 ninth, Lee dropped her tee shot inside five feet and drained the putt for another birdie to go out in 29 on the par-70 layout.
Lee picked up another eagle at the par-5 13th and played her approach to 12 feet for a birdie at the par-4 15th.
The 25-year-old kept on rolling and landed her second shot inside seven feet at the par-4 16th. Lee then sank the putt for another birdie to reach 10 under.
With two holes remaining, Lee had more than an excellent chance of breaking 60 and joining Annika Sorenstam as the only women to shoot 59 on the LPGA Tour, even though she didn't know it.
'My caddie told me,' said the Korean. 'After I finished the 18th hole.'
Lee missed the putting surface at the par-3 17th, but still had a shot at birdie. She putted from off the green and watched as the ball came within inches of the cup.
At the closing par-four, Lee's approach shot stopped within 14 feet of the hole, but this time her birdie try missed left.
'I worked a lot this winter,' said Lee, whose best finish on the LPGA Tour was a tie for second at the 2003 Longs Drugs Challenge. 'So it should be good for this year.'
Lee is not the first to flirt with the magical number at this event in recent memory. Last year, Meg Mallon shot a 60 of her own in the second round.
Stupples collected three birdies and a bogey over her first nine holes to make the turn at 2 under.
She picked up steam on the inward half, however, with three straight birdies starting at the par-3 11th.
Stupples added a birdie at the 15th and played her second shot to six feet for a birdie at the last.
'I hit the ball with the wedges, and that was pretty good today,' said Stupples. 'I gave myself opportunities from that. And when I got into trouble, I got out of it pretty well.'
Jackie Gallagher-Smith, Eva Dahllof and Stacy Prammanasudh tied for fifth place at 4-under-par 66. Se Ri Pak was one shot further back along with Mi Hyun Kim, Riko Higashio and Laurie Rinker at 3-under-par 67.
Defending champion Wendy Doolan struggled on Thursday with an opening-round 74.