Pieters drained a 30-foot birdie putt on the 18th hole to cap a 4-under-par 66. Johnson birdied two of his last three holes for a 65 that gave him a 54-hole total of 9-under 201.
Alone in third place one stroke behind the leaders is Scott Hend, who shot a 63 for the low round of the day.
Hideki Matsuyama of Japan was by himself in fourth, two behind the leaders, after a second consecutive 67. Tied for fifth were Adam Hadwin (67) of Canada, Charley Hoffman (67) and Rory McIlroy (68) of Northern Ireland.
Pieters, who is from Belgium and played college golf at the University of Illinois, made five early birdies to vault into the lead after eight holes, but he stalled from there and parred 11 holes the rest of the way.
"My ball striking was good. My driver was not good," Pieters said. "It's been like that for three days and I've managed to score really well. I'm pretty happy. I'm in a good position thus far."
Johnson is bidding for his first victory since he won the British Open in 2015. He capped the third round by making a 10-foot birdie putt on the 18th hole in a round that included eight birdies and three bogeys.
"It was good," Johnson said of his round. "I have myself a lot of opportunities.Just a sold day all around. I drove it good. I putted good. I felt like I left some out there, but at the same time I know I made a few too.
"There's still 18 holes left. I still have to keep the pedal down."
Hend began the day with a birdie at No. 1 and added three birdies on the front nine along with one bogey. After the turn, he opened with a birdie at No. 10 and added three more at Nos. 14, 15 and 18. The Australian nailed a 15-foot putt on the last hole to cap his 7-under round.
McIlroy was unable to make a big move Saturday with four birdies offset by two bogeys and par golf on the final six holes. But he is pleased to be within striking distance going into the final round.
"I did leave a few out there," McIlroy said. "A little bit frustrating, but I'm only three behind. It's just a matter of holing the putts and getting some momentum going. I gave myself plenty of chances on the back nine. I just didn't make as many as I wanted to."