Kim won with a tap-in par on the par-4 18th on the second extra hole at Club de Golf Mexico.
''This is the greatest win in my life, without a doubt,'' Kim said.
The 30-year-old American celebrated with an exuberant dance and jumped up and down on the green. Kim hugged caddie T.J. Jones, boyfriend Duncan French and friend Michelle Wie, with Wie wiping away tears after embracing the winner. French is Wie's caddie.
''I can't put into words how I'm feeling right now. Kim said. ''I'm still riding so many emotions. It feels surreal. I can't explain. I'm so overwhelmed right now. I'm so proud and honored to be a champion for such an incredible tournament.''
Kim won for the first time since the Tournament of Champions in November 2005, ending a 221-tournament victory drought. She also won the 2004 Longs Drug Challenge.
''She jumped up and she was really happy,'' Feng said. ''I felt the same thing. I was really happy for her because she's a good player, good person.''
Feng, from China, drove left on the second playoff hole, with her ball settling too close to a tree truck for her to do anything but chip sideways into the fairway. With Kim already pin high 20 feet right of the hole, Feng hit her 9-iron approach to 25 feet - also to the right. Her par putt slid by the right edge, and Kim two-putted for the emotional victory.
Feng had a birdie putt to win on No. 18 on the first playoff hole, but left her 15-footer inches short and left. Kim then made a 5-foot par putt to extend the playoff.
Kim bogeyed the final hole of regulation - running a 3-footer past the left edge - for a 1-under 71 that left her tied with Feng at 15-under 273.
''My brain just didn't work right there,'' Kim said.
Feng shot 66.
''I thought other than Christina, the rest of us were competing for second place,'' Feng said. ''I never thought I could catch up. ... I had a really solid run.''
After bogeying Nos. 14 and 15 to fall into a tie, Kim rebounded with birdies on Nos. 16 and 17 to take a one-stroke lead.
''I was just thinking that there were still chances for me to make birdie,'' Kim said. ''All the last four days I was thinking, 'What would Lorena do?' She would gather herself. She would know that she has the ability to make more birdies and just play her best. And whatever happens, happens.''
Feng played the four par 5s in 5 under, making an eagle on No. 2 and birdies on Nos. 6, 11 and 17, and chipped in for birdie on the par-4 15th. She won the LPGA Malaysia last month.
''She shot 6 under. She played lights out,'' Kim said. ''It was a very easy round to watch her play. She just made it look so easy while I was bumbling around with the trees.''
Lexi Thompson, the winner last year at Guadalajara Country Club, holed out from 129 yards for a double eagle on No. 2. She finished with a 73, also making two double bogeys, five bogeys and five birdies, to tie for 15th in the 36-player field at 3 under.