Park picked up seven shots in her last 12 holes for a 7-under 65 to reel in Ko, who lost the lead at Turnberry when her second shot at No. 16 rolled into the burn. Park finished on 12-under 276.
The LPGA Tour is calling Park's achievement a career Grand Slam, although Park hasn't won the Evian Championship since it was given the status of a fifth major in 2013. She did win Evian in 2012.
"This is something I have been dreaming of all my life and all my career," Park said. "To finally reach my goal is just a relief. ... I don't know what else to go for."
The Evian is staged next month in France, when Park can complete the modern-day Grand Slam and end the debate.
Park won her seventh major overall, and has captured six of the last 14.
Louise Suggs, Mickey Wright, Pat Bradley, Juli Inkster, Karrie Webb and Annika Sorenstam are the other women to have won four different majors.
Ko, a precocious 20-year-old playing outside Asia for the first time and in her first links tournament, finished runner-up on 9 under after a 71.
Ko began the final round tied for the lead with Teresa Lu, and made an eagle and two birdies around the turn to go three shots clear after 10 holes.
Park pulled away from a chasing pack that included No. 2-ranked Lydia Ko and Suzann Pettersen at the start of the back nine, making Ko -- her protg -- work for the win.
As Ko began to tighten, Park made her move. She rolled in an eagle putt from 20 feet at No. 14 to close to within one shot, Ko missed a par putt on No. 13 soon after and Park holed a 4-footer for birdie at No. 16 to take the lead for the first time this championship.
Ko's chances of reclaiming the lead virtually ended when she hit her approach shot on the par-4 No. 16 right and into a stream.
The 18-year-old Lydia Ko, bidding to become the youngest major winner, shot 69 and tied for third on 8 under with So Yeon Ryu (68).
Inbee Park makes history with British Open victory
Top-ranked Inbee Park rallies in the final round to win the Women's British Open, her fourth different major title.