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The second-ranked American had led by four shots after his third straight birdie at the sixth hole at East Lake in Atlanta, Georgia.
But he fell a stroke behind Chappell with a double-bogey at the 17th before regaining a share of the lead with a birdie at the par-five finishing hole.
Chappell, meanwhile, was a model of consistency, and patience, with two birdies in his two-under 68 that left him level with Johnson eight-under par 202.
They were two strokes in front of Northern Ireland star Rory McIlroy and American Ryan Moore, who both carded four-under par 66 for 204.
Japan's Hideki Matsuyama was alone on 205 after a 68. Jason Dufner was a further stroke back, while South African Charl Schwartzel, who had back-to-back eagles in his 66 was tied with England's Paul Casey on 207.
Johnson, who started the day with a one-stroke edge over Chappell, wasted no time in stretching his lead.
He slowed with a bogey at the seventh, and a birdie at 12 was followed by back-to-back bogeys at 13 and 14.
Johnson rolled in a 15-foot birdie putt at the par-three 15th, but his double bogey at 17, where he was twice in the rough and in a bunker, saw Chappell take the lead with a 10-foot birdie putt at the same hole.
"I thought I played pretty well, just a couple of loose drives on the back side," Johnson said. "Other than that, I thought the round was pretty solid.
"I missed a few good opportunities for birdie on 8, 9 and 2, but otherwise, I felt like I hit good putts," he said. "They just didn't go in."
Chappell had just two birdies all day. He'd ground out 11 straight pars before chipping in for birdie at the 12th - a result that brought an emotional fist-pump from the American seeking his first US tour title.
"Probably the most difficult round I played this week in terms of ball control and positions that I was in," Chappell said.
"But I fought real hard and the chip-in at 12 kind of got me fired up a little bit. I think that was the glue that kind of kept the round together."
Johnson, in contrast, is the hottest player on the tour right now, with three wins since June including his breakthrough major title at the US Open.
Johnson leads the FedExCup playoff standings, with a $10 million bonus on offer to the playoff points winner on Sunday.
The prize is Johnson's if he wins or finishes alone in second, but he's not the only one in contention.
McIlroy still has a shot, if he can win his first Tour Championship crown and Johnson falters.
"First and foremost I need to go out tomorrow and try to win the golf tournament," McIlroy said.
Moore, also in the FedExCup hunt, said it would be important to focus on golf and put the possible playoff scenarios out of his mind.
"Of all the days to just kind of focus and take it one shot at a time and play a round of golf, tomorrow's the day," he said. "It's going to be hard to do with everything that's on the line right now, but I like where my game's at."
Third-round scores on Saturday in the $8.5 million PGA Tour Championship, final event in the FedExCup playoffs at East Lake in Atlanta, Georgia (USA unless noted, par-70):
202 - Dustin Johnson 66-67-69, Kevin Chappell 66-68-68
204 - Rory McIlroy (NIR) 68-70-66, Ryan Moore 70-68-66
205 - Hideki Matsuyama (JPN) 66-71-68
206 - Jason Dufner 73-67-66
207 - Charl Schwartzel (RSA) 74-67-66, Paul Casey (ENG) 68-70-69
208 - Justin Thomas 68-71-69
209 - Emiliano Grillo (ARG) 73-70-66
210 - Roberto Castro 73-70-67, Matt Kuchar 69-73-68, Russell Knox (SCO) 73-66-71
211 - Bubba Watson 72-73-66, Daniel Berger 74-69-68, Gary Woodland 72-70-69, Adam Scott (AUS) 69-71-71
212 - William McGirt 77-69-66, Jordan Spieth 68-72-72
213 - Patrick Reed 73-70-70, Brandt Snedeker 72-69-72, Kim Si-Woo (KOR) 67-72-74, Kevin Kisner 67-70-76
214 - Jhonattan Vegas (VEN) 70-73-71, J.B. Holmes 73-71-70
215 - Phil Mickelson 74-72-69, Sean O'Hair 74-68-73
219 - Jimmy Walker 74-74-71
223 - Kevin Na 77-74-72
Jason Day (AUS) 67-WD
