Woods was 1-up through four holes when Clark, a South African, birdied the fifth to square the match.
Clark went 1-up when he parred the par-3, 185-yard sixth hole and Woods bogeyed. Woods left his tee shot in a greenside bunker, blasted out to 15 feet and missed the putt for par.
Drawing a huge gallery on a 77-degree afternoon, Woods birdied the par-5, 574-yard second hole to go 1-up over Clark.
Woods and Clark both parred the par-4, 460-yard hole. On the second, Woods hit his second shot within 28 feet, then left his putt for eagle a few feet short. Clark conceded Woods' birdie after missing his own 70-footer for birdie.
The main question was how Woods' reconstructed left knee would respond a day after his first competitive round since winning the U.S. Open eight months ago.
Two years ago, Woods defeated Clark 5 and 4.
Woods' return generated worldwide attention as fans wondered whether he would be the same player who's won 65 times on the PGA Tour, including 14 majors.
He quickly dispelled doubts in the opening round, dispatching game Australian Brendan Jones 3 and 2 on Wednesday. Afterward, Woods made it clear he didn't come to the Arizona desert to enjoy the 80-degree temperatures.
"I don't go to an event that I don't think I can win," Woods said. "Why go? It doesn't make any sense to me. So I entered this event with the same intention I do every event since I was a little boy, and that's to win."
The 64-man field was pared to 32 for the second round.
In an early match Thursday, former Masters champions Phil Mickelson was 2-up over Zach Johnson with two holes to play.