The Brit, whose only previous WTA crown came at Nottingham last June, moves into the world rankings top 30 for the first time in her career.
Boulter had upset third seed Emma Navarro of the United States in the semi-finals.
Kostyuk, ranked 34th, had beaten top-seeded Jessica Pegula to reach the final but struggled after winning the first set against the big-hitting Briton.
After rain forced a delay in the final, Boulter handed the hard-fought first set to the Ukrainian on her seventh set point after she double faulted.
But having forced her way back from 5-2 down to 5-5 in that set, the momentum was with Boulter and she took that into the second set.
A fine forehand winner down the line put her 3-1 ahead and she broke again to lead 5-2 on the way to forcing a third set.
As Kostyuk tired, Boulter grew in confidence, breaking her opponent twice before firing two emphatic aces in the final game.
Boulter converted seven of her 11 break points and saved seven of the 11 that she faced in a win that will move her to 27th in the world rankings.
"I really didn't think I would be standing here with a trophy right now, I really didn't," said Boulter, who was cheered on by her Australian boyfriend Alex de Minaur, who had flown in after winning the ATP title in Acapulco on Saturday.
