McIlroy bogeyed three of the first eight holes but made the turn at level par after a 34-foot birdie putt on the ninth.
The 29-year-old then closed strongly to salvage a round of 68 with four birdies and one bogey on the back nine.
That was not enough to keep pace with Johnson though, despite the American making a double bogey on the par-four 10th.
Johnson eagled the second hole after a superb tee shot and made the turn in 32. He then bounced back from his setback on the 10th with four birdies in the last eight holes.
Patrick Reed had two eagles in a round of 64 that lifted him up 17 places and into a four-way share of third, seven shots back, alongside Patrick Cantlay, Sergio Garcia and Cameron Smith.
Tiger Woods' round fell apart late on as he could only manage a 70, dropping down into a share of ninth place.
Woods was on course for much better until he four-putted the par-five 15th for a double-bogey and dropped another shot on the 16th, though he birdied the 18th to limit the damage.
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies