Johnson, the 2007 Masters champion, denied the world number one a sixth victory in the unofficial tournament he hosts for the benefit of his charitable foundation.
After appearing to drown his chances by hitting his second shot into the water at the final hole of regulation, Johnson holed his shot from the drop area 65 yards out to make par and complete a four-under 68.
Woods parred the 72nd hole from a greenside bunker to cap a two-under 70 that left him tied with Johnson on 13-under 275.
Johnson parred the first playoff hole as Woods — in the bunker again — missed a five-footer to extend the playoff.
Woods started the day with a two-shot lead over Johnson and led by four after Johnson bogeyed the 10th.
Johnson bounced back with birdies at 11 and 12, and when Woods bogeyed 14 the lead was down to one.
After both birdied 16, Johnson birdied 17 to send them to the 18th tee tied for the lead.
Johnson was safely in the fairway as Woods was in the rough off the tee. From a difficult hillside lie he found a bunker below the elevated green.
Johnson, however, then hit into the water. He could only laugh when his shot from the drop area took a couple of small hops and spun into the cup.