Begu defeated Vekic 6-4, 6-4 behind a solid service game. She landed three-quarters of her first serves and won 70 percent of the points on serve. Vekic managed just one break point, which she converted, against Begu.
Begu cashed in her only break-point chance of the first set, when she dropped just four points on serve, and had two more breaks in the second set.
It was her third trip to the finals of a WTA event. She beat three seeded players, including second-seeded Urszula Radwanska, to get to the title match.
Vekic, 16, is the youngest player to make a WTA final since 15-year-old Tamira Paszek won in Slovenia in 2006.
Begu picked up $37,000 along with her first tournament title. Vekic claimed $19,000 as runner-up.