Jordan Spieth - still searching for his first win of the year - managed a five-under 66 to move into seven under, enough to leave him tied for 20th.
Defending champion Billy Horschel, who started on the back nine, had an eagle at the par-five 14th en route to a three-under 68 that left him seven shots back.
Instead, Leishman opened with an eagle, started the back nine with three straight birdies and reached nine under with another eagle at the 14th.
The 34-year-old Victorian, a three-time victor on the PGA Tour, started with an eagle on the par-5 first hole and never looked back, claiming another eagle at the 14th to go with six birdies.
The 24-year-old had three birdies and three bogeys and is in danger of going another year without surpassing his best Nelson finish - a tie for 16th when he was a 16-year-old high school junior in 2010.
South African Branden Grace and Americans JJ Spaun and Keith Mitchell shared third on 265.
"When you shoot a really low one like I did yesterday it's hard to back it up", said Leishman, who had a 15-under-par total of 127 and a two-shot lead with the afternoon starters still on the course. He shot a bogey-free 65 and is among five players at 10 under a week after finishing tied for 11th at the Players Championship.
GETTY IMAGES Marc Leishman shot his lowest round on the PGA Tour
Speaking after the round to The Golf Channel, Leishman said: "I didn't even think about 59, every other time I have been close to it I have thought about it".
Leishman opened with a 61 in Trinity Forest's debut, after the Nelson spent 35 years at the TPC Four Seasons. "I know better than that on certain shots". "I didn't have quite as many drivers as I had the last couple of days because it was firmer".
Maybe part of the reason Kuch didn't play well is that he just doesn't like the golf course.
"This will be the second Saturday in a row I'm in the final group", Wise said.
Maybe Spieth isn't the only one who feels at home.
"I'm reading the greens well and hitting the ball well".
"It was about as easy tee to green as I've ever seen this place", Spieth said.
"I finish 16th tomorrow and people are going to say, 'What's wrong?'" said Spieth, a three-time major victor.