The 24-year-old from California two-putted for a par five at the first extra hole to defeat former British Open champion Jeong Jang of South Korea.
Like Anna Kournikova in tennis, Solheim Cup star Gulbis had become more famous for her calendars and reality tevelvision shows than her sporting success, but she put that right in the French town of Evian.
Gulbis finished the tournament on four under par after a stellar final round of two-under 70, including three birdies on the outward nine, but Jang matched her score to force extra holes.
"There has been a lot of hype in America with people asking if I was ever going to win," admitted Gulbis.
"This is my sixth year on tour but I knew if I was patient it would eventually happen. But it does feels great to have won at last."
Mexico's world number one Lorena Ochoa (68), South Korea's Shin Ji-Yai and 47-year-old American Inkster, who had a 75 to lose her chance of becoming the Tour's oldest winner, tied for third on three under.
