The 27-year-old, whose only previous win on the tour came at the South Africa Open in January, carded a superb six-under-par final round of 66 to claim victory by two strokes.
On a good day for English golfers, David Howell and Anthony Wall finished in a five-strong group tied for second and also sealed their places at the Open at St Andrews in July.
“It’s unbelievable,” said Sullivan. “I never imagined it would happen again so quickly.”
Teenage women’s world No1 Lydia Ko continued her superb run by winning the New Zealand Open on home soil yesterday. The 17-year-old followed up her victory at the Australian Open seven days earlier with a four-shot win.