Champions Tour rookie Craig Stadler notched a 6-under-par 66. He shares fourth place with Tom Jenkins and James Mason.
Fleisher opened with a birdie at the first on the Jones Course at Rock Barn Golf Club and Spa. He birdied both front-nine par-5s, Nos. 4 and 7 to move to minus-3.
Around the turn, the Verizon Classic winner stumbled to a three-putt bogey at the 10th. He came right back to sink a 20-footer for birdie at the next to atone for that mistake.
Fleisher caught fire from there. He rolled in a birdie putt at the par-4 13th and followed with kick-in birdie at No. 14. Fleisher made it three straight when he birdied the 15th after dropping an 8-iron within two feet.
The 54-year-old continued his roll on No. 16 when he sank a birdie putt. Fleisher capped a remarkable run of five straight birdies when he made a 10-footer for birdie on the 17th.
'What is it about North Carolina? If I knew I would write a book and I'd sell it for a fortune,' said Fleisher, who has won four straight years in the state of North Carolina. 'I just like the surroundings.'
Fleisher, who was paired with Nelson and Stadler, used the strong play in the group to keep him going.
'We all played off each other out there,' said Fleisher. 'It would have been nice to birdie the last hole. Nelson is playing very good golf and he is confident.'
Nelson, like Fleisher, opened with a birdie on the first. He later sank back-to-back birdies from the fourth to climb to 3-under. However, he faltered to a bogey on the seventh but came right back to birdie the eighth to make the turn at minus-3.
On the back side, Nelson made a six-foot birdie on the 12th and made it two straight with a birdie on No. 13.
Nelson, who won the Constellation Energy Classic earlier this season, then holed a lob-wedge from 80 yards out for eagle at the 14th to get to 7-under. He finished his hot streak with a birdie at No. 15 to grab a share of the lead.
'I wanted to shoot 63 very badly,' said Nelson, who missed a short birdie try on the 18th. 'Seeing guys at 5-under early helped the guys who were playing later. We said, `Well, maybe we can go low here.''
Nelson, who has won 17 times in six years on the Champions Tour, knows this may be one of his last chances to win.
'You don't know when age is going to come and catch up with you, but I feel as good right now as I've felt ever since I've been out there,' said Nelson. 'I've always played fairly well when I've been healthy and right now I feel pretty good.'
Dana Quigley, starting his 229th consecutive event for which he's been eligible on the Champions Tour, carded a 5-under-par 67. He is joined in seventh place by Gil Morgan, Doug Tewell, Jim Thorpe and Jay Sigel, who played his college golf at nearby Wake Forest.
Bobby Wadkins and his brother, Lanny, lead a group of seven players at 4-under-par 68.