It was Kim's second LPGA victory this season and the seventh of her career. The runner-up finish extended Gulbis' winless drought to 123 starts on the LPGA Tour.
Kim came from four shots back to tie Gulbis for the lead, making three birdies on the back nine in her final round.
Gulbis, who played the front nine at 5-under 29, parred her final eight holes to tie Kim at 18-under 266 for the tournament.
Both players shot 6-under 66 in the final round to set up the sudden-death payoff.
Kim won the Ginn Clubs & Resorts Open in April and has seven other top-10 finishes this season, outside of her two victories. She earned $180,000 for the win Sunday.
Celebrity Couples of 2014 [PHOTOS]