Woods -- a five-time winner of the event -- also looked poised to shoot 70 when he came to the 18th green with a three-foot putt for birdie. But he missed the putt and settled for a one-under 71.
Johnson, who finished second in the World Challenge in 2007 and 2011, had seven birdies and two bogeys in Thursday's round.
One of his birdies came on the par-four 14th hole when he sank a 25-foot putt that had to travel up over a ridge.
"It was a good start," Johnson said. "Just a real solid day all around."
Two-time and defending World Challenge winner