The 18-year-old rattled in two long putts in the final two holes to card a six-under 65 to tie with Ko, who recorded a final-round 67.
Henderson held her nerve to sink a three-foot putt at the first extra hole to win the £370,000 first prize.
Scotland's Catriona Matthew was tied for 12th after a final round 72.
The 46-year-old had been just three shots off the lead going into the final round.
New Zealander Ko had led by one overnight at the Sahalee Country Club as she sought to win her third consecutive major championship.
She continued to be in front for most of the final round before Henderson made a late move.
Henderson said the win felt like a dream.
"I don't know whether I thought it would come true," she said. "I'm just speechless. I don't know what's going on in my head right now."
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Canada's Brooke Henderson claimed her first win in a major on Sunday, beating world number one Lydia Ko in a play-off at the PGA Championship in Seattle.