Johnson briefly got to nine under with four consecutive birdies on his back nine but he bogeyed the par-four 17th after hitting his second shot into a greenside bunker before posting an eight-under 64, one ahead of compatriot Brendan Steele.
Australian world number one Jason Day was among a group of five players who opened with 66s, second-ranked American Spieth carded a 70 and third-ranked Northern Irishman McIlroy mixed six birdies with three bogeys and a double to return a 71.
Johnson, a nine-times winner on the PGA Tour whose best finish this season was a third place at the Houston Open in April, set the tone with an explosive birdie-birdie-birdie start.
"I just played well right out of the gates," Johnson, 31, told reporters. "I hit great shots in the right spot on the greens and rolled in some putts.
"This year, I've felt like I'm playing well. I just haven't quite played up to my potential. With me, it has everything to do with the putter. I rolled it well today."
Ohio-based Day, who clinched his third PGA Tour victory of the season at the Players Championship last month, was delighted to make a good start at a venue where he has struggled in the past.
"I hit it nice from tee to green, gave myself opportunities on the greens and then also capitalised on those," he said. "I'm pleased because usually I come into this event and start pretty poorly, maybe put a little too much pressure on myself.
"It is my home course, and you want to play well, and you want to win this tournament because of Mr. Nicklaus and what he's done."
Spieth, who landed his eighth PGA Tour career win at the Colonial tournament on Sunday, relied on his usually brilliant putting to offset an iffy display with his approach shots.
"I felt awesome on the greens," Spieth said after a round that included five birdies and three bogeys. "Just missed a lot of greens.
"I didn't have great control of the ball today, but I think it's close. My putter made up for it for the most part."
McIlroy, like Spieth a tournament winner in his most recent start after clinching the European Tour's Irish Open 11 days ago, salvaged an erratic round with a birdie-birdie finish.
"I felt like I hit it well enough off the tee to get it close enough to the pins, and most of the time I did that," said the Northern Irishman. "But I just hit a few loose shots, and that's what cost me today."
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Andrew Both/Steve Keating)
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