The U.S. Open champion posted a 10-under 132 halfway total in the event formerly known as the Los Angeles Open, one stroke better than fellow Americans Pat Perez (66) and Cameron Tringale (64).
Venezuelan Jhonattan Vegas trailed by three shots along with Americans Patrick Rodgers and J.T. Poston, with former world number one Luke Donald of England among a group another stroke back.
The second round was completed nearly 24 hours behind schedule, a result of Friday's near washout as a major winter storm lashed southern California with torrential rain and strong winds.
Seventy-one players made the cut, which fell at even par, before the third round started late afternoon, though the leaders will not tee off until Sunday morning.
Officials still hope to complete the scheduled 72 holes by late Sunday, which should be do-able given the decent forecast.
World number three Johnson will jump to top spot if he wins on Sunday, as long as current number one Jason Day finishes lower than third.
Australian Day faces a tall order to do so, eight strokes off the halfway pace in a tie for 40th.
Johnson is feeling at home on a course where he twice has finished runner-up.
"I'm very comfortable," he told Golf Channel. "I’ve had a lot of good finishes so it gives me a lot of confidence going into the last two rounds."
His approach game went off the boil mid-round but he used a deft short touch, assisted by the soft conditions, to keep his card blemish-free.
"I hit a few squirrelly shots," he admitted. "I drove it well all day. I just hit some poor iron shots in the middle of the round. Didn’t make any of my short (putts) but made a few long ones."
Those long ones included a 35-footer at the 15th and a 20-footer at the last.
Tringale, meanwhile, finished in style, slam-dunking an 80-yard wedge shot to birdie the last and keep the heat on Johnson.
Perez also closed well with birdies at the final two holes.
World number five Hideki Matsuyama of Japan, who also started the week with a chance of becoming number one, missed his first cut since August, shooting 80 for six-over 148.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Gene Cherry)
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