Third-seed Ostapenko had appeared on course for a routine victory after winning the first set of the tournament opener, but world No. 317 Kucova rallied to take the game to a decider.
The Slovakian was without a win over a top 20 player in two years, yet showed no signs of nerves to comfortably take the third set and send her more illustrious opponent home early.
Naomi Osaka, the 2018 US Open Champion and set to be the star name in Hong Kong, withdrew from the tournament citing lower back pain.
Despite the injury, the 20-year old still travelled to Hong Kong to meet with fans.