Australia's Rodger Davis shot a 65 of his own to finish two shots off the pace at 9-under-par 135 along with Jim Thorpe.
Barr earned a breakthrough victory at last week's Royal Caribbean Golf Classic. He became the first Canadian to win on the Champions Tour and has put himself in position again thanks to a brilliant performance at TwinEagles Golf Club.
'If I come out tomorrow and beat the golf course, that's all I can do,' said Barr. 'Hopefully I'll be in a good frame of mind to go shoot a good round. Last week might have helped me a little bit. I'm a bit more relaxed.'
He hit a pitching-wedge to seven feet for birdie at the par-5 third. Barr then birdied the fifth and chipped his third shot to 12 feet for a birdie at the sixth.
Barr stumbled with a three-putt bogey at the 10th but responded with a birdie at the 12th. At the par-5 13th, Barr hit a 3-wood to 15 feet and drained the putt for an eagle to move to minus-9.
The 50-year-old dropped a 9-iron within 10 feet for birdie at the 16th and finished his round with a birdie at the last to take the outright lead.
'I struck the ball quite well tee to green. Overall an excellent putting round and I think I only missed two greens and those were on the fringes,' said Barr, a two-time winner on the PGA Tour. 'I'm just riding the wave right now.'
Fernandez, who held the lead after the first round, got started with a birdie at the third. He birdied the 10th and added another birdie at 13. At the par-5 17th, Fernandez hit his third shot to four feet for birdie and a flawless 68.
'Today was a little bit more tricky. The wind switched and it was harder to get it close,' said Fernandez. 'Overall, I'm pleased with the score.'
Jay Overton was a Monday qualifier for this event and managed to put his name in the books on Saturday. The 52-year-old played a bogey-free round with nine birdies for a course-record 63.
'I felt like I belonged in the tournament and just had to go out and prove it on Monday,' said Overton. 'For a Monday qualifier, this is a nine-day work week.'
Tom Watson, who owned the previous course record, carded a 68 to finish in a tie for sixth with Mike McCullough at 8-under-par 136.
Bruce Lietzke, Jim Ahern, Doug Tewell, Tom Purtzer, Tom Jenkins and Jim Dent finished tied for eighth at 7-under-par 137. Walter Hall, Mike Smith, Isao Aoki and Gil Morgan were one shot further back at 6-under-par 138.
Defending champion Hale Irwin had three birdies and one bogey for a round of 2-under 70. Irwin finished alongside Q-School medalist Des Smyth, Larry Nelson, Jose Maria Canizares, Leonard Thompson, Gary Koch and Bobby Walzel at 5-under-par 139.
Jack Nicklaus, the architect of this venue, recovered after a disappointing opening round on his 'home turf.' Nicklaus was 4-under on his round with two holes to play on Saturday but double-bogeyed the par-4 eighth and bogeyed the par-4 ninth for a round of 71. Nicklaus finished tied for 59th at 3-over-par 147.