England's Tommy Fleetwood is tied for second place with American's Pat Perez and Xander Schauffele at 5-under-par.
Johnson, 42, was the clubhouse leader after he carded a 4-under-par 67 in his morning round. He had a rough day at the start, making bogey on hole No. 1. But Johnson got hot after making par on No. 2. The No. 52 golfer in the world followed the par by making birdies on three of the next four holes and carded a 2-under-par 34 on the front nine. Johnson made par on seven of the final nine holes. He hit a birdie on No. 14 to earn a share of the lead. He then birdied No. 18 to earn the outright lead, before Kisner's round was completed.
The birdie came on a very long putt on the par-4 18th green. But Johnson wouldn't be alone for long at 6-under-par.
Kisner -- who led the field at 5-under-par entering the round -- made par on the first four holes before heating up. He birdied No. 5, No. 7 and No. 9, to hand in a 2-under-par 34 on the front nine, leading Johnson by one stroke. But Kisner lost strokes down the stretch, carding a 36 on the back nine, which included a bogey and a double bogey.
Johnson has two major championships during his professional tenure, including the 2007 Masters and the 2015 British Open. Kisner has never claimed a victory in a major.
Rory McIlroy, Erik van Rooyen, Matt Kuchar, Tony Finau and Zander Lombard are tied at 4-under-par and sit in sixth place. Kevin Chappell, Jordan Spieth and Rickie Fowler are tied for 11th at 3-under-par.
Kisner and Johnson tee off at 11 a.m. Saturday. McIlroy and Schauffele tee off at 10:40 a.m. while Spieth and Chappell tee off at 10:10 a.m. Woods tees off at 8:15 a.m. with Shaun Norris.