Grace Park and Young-A Yang finished three shots off the pace at 7-under-par 206. Karen Stupples and Jeong Jang followed at 6-under-par 207.
Steinhauer was six shots off the lead to start the day at Wykagyl Country Club and parred her first seven holes before dropping a 9-iron within 12 feet of the cup for a birdie at the par-4 eighth.
The 41-year-old hit a 7-iron inside 10 feet for a birdie at the par-4 11th and reached the par-5 15th in two. Steinhauer was unable to convert for eagle but she two-putted for birdie to move to minus-8.
At the par-5 18th, Steinhauer again found the green with a 3-wood, but this time her second shot stopped 5 feet from the cup.
'There is nothing like hitting a golf shot that you picture in your mind before,' said Steinhauer. 'I was kind of pacing around because I could just picture the shot, and I couldn't wait to hit it and got up over it and made a wonderful swing. It was a great feeling.'
She then drained the short eagle try to match Morgan in the lead after 54 holes.
'It's been a long, long time,' said Steinhauer, whose last victory on the LPGA Tour came at the 1999 Women's British Open. 'I have been playing better and feeling really good about where I am. I don't have a lot of expectations because it's been a while since I've been here.'
Morgan, who fired a 64 on Friday to take the 36-hole lead on the LPGA Tour for the first time in her career, had some trouble on the course on Saturday.
She picked up a birdie at the third but bogeyed the par-3 seventh after she missed the green off the tee. Morgan's struggles continued on the back half, where she dropped shots with bogeys at the 12th and the 14th.
'I just had a couple of bad swings on the back side,' said Morgan. 'A couple of my not favorite holes.'
Morgan salvaged what she could with a birdie at the 15th on her way to a round of 72.
'I thought there would be a few more people ahead, playing better than they did, but I kind of hung in there,' said Morgan, who is looking for her maiden title on the LPGA Tour. 'I hit a few bad shots, and didn't really make any putts. So I got to work on it and hope it's better tomorrow.'
Park, who won this year's Kraft Nabisco Championship, collected four birdies and four bogeys for a round of 71.
'I'm just glad that the round is over,' said Park. 'I got a little frustrated out there.'
Kelli Kuehne carded a 66 to climb into a share of seventh place along with Young Kim, Amy Hung and Nadina Taylor at 4-under-par 209.
Silvia Cavalleri was one shot further back at 3-under-par 210.