South African-born Wise, who started hours after Leishman, came close to catching the Australian with an eight-under 63.
Six birdies and an eagle put the 21-year-old at 14-under 128, a stroke ahead of American Brian Gay, whose nine-under 62 was the low round of the day.
Americans Jimmy Walker (67), Eric Axley (65) and Kevin Nau (65) shared fourth on 131, while former world number one Adam Scott of Australia (65) was in a group of five players one shot further back.
Leishman found the links-style Trinity Forest, the tournament's new home, much to his liking.
"I enjoy this type of golf," the world number 16 told reporters. "It reminds me of back in Australia.
"I like how the ball is running out on the fairways. I'm reading the greens well and I'm hitting the ball well."
Leishman said there was a big difference in him shooting a four-over 76 in the final round of last week's Players Championship and his play in Dallas.
"My speed has been really great," he told the Golf Channel. "Last week at The Players I hit a lot of really good putts that lipped out, and this week they're going in.
"Some weeks you read them really well, and some weeks you're not sure if a 10-footer is slightly uphill or downhill, and that affects your speed and line and everything."
Wise, who tied for second in the Wells Fargo Championship two weeks ago, said he used the Australian as an incentive.
"I saw Marc Leishman way up there at 15-under and I started the day at six (under) so I knew I had a long ways to go," Wise told PGA Tour Radio.
"It was nice when I started getting it going to see his numbers still ahead of me so I could keep chasing and chasing. Even though I was playing tough, I felt like I had to go get him a little bit, which was fun."
Gay had as many birdies as Leishman on Thursday, 10, but a bogey at the par-four fourth spoiled his chances of equalling Leishman's round.
Six of his birdies came on the back nine.
"My putting has been really down the last month or two, and that's my strong suit, so I have just been waiting for it to turn around," Gay admitted.
Hometown favourite Jordan Spieth found improvement with a solid 66 but was still eight strokes off the pace on a low-scoring day.
(Reporting by Gene Cherry in Raleigh, North Carolina; Editing by John O'Brien)
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