Kostyuk had looked out for the count after the first set and Sakkari's flawless power game but recovered after a lengthy rain interruption in the first set and another in the second to turn the match around.
The match had started also with a brief rain interruption but the Ukrainian still managed to earn her first win over her opponent at the third attempt.
"Probably I have never been interrupted twice," she said of the two rain suspensions and the brief interruption early in the first set.
"Every time I was going into a rain break I was going into different emotions," said Kostyuk, who was also emotional several times during the match.
"I was crying during the breaks. I tried to figure out what was happening. I tried to come back into the match. It was not easy.
"I just tried to fight and want to say thank you to my team," she said.
Sakkari made a flawless start and bagged two breaks to move 5-0 up with Kostyuk's error count rising sharply, but the 28th-ranked Ukrainian got some respite when rain stopped play.
Sakkari initially looked undisturbed by the rain suspension, taking the first set soon after the restart.
Kostyuk, however, turned the tables in the second set with aggressive play that forced Sakkari into more unforced errors.
She broke the Greek twice to go 5-3 up before forcing a decider.
Sakkari was wilting and Kostyuk earned nine consecutive games en route to a 5-1 lead and sealed victory on her first match point.
