The duo finished the regulation 72 holes on 13-under-par 275, world number one Woods parring the last for a two-under 70 and Johnson, who birdied four of the last eight holes, sensationally closing with a 68.
Tied for the lead at 13 under, Johnson appeared to have the title firmly in his grasp when five-times champion Woods, from the left rough at the 18th, dumped his approach into a greenside bunker.
However Johnson, perfectly positioned in the fairway, stunningly found water with his second, took a penalty drop and then holed out from 65 yards in the fairway for a par four as the galleries erupted in deafening celebration.
Woods did well to get up and down from the bunker to take the tournament into extra holes.
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