The birth is a "true rarity" as their mother is the oldest giant panda on record to give birth for the first time.
Ying Ying delivered the cubs on Thursday, one day before her 19th birthday. If she were a human, she would be the equivalent of a 57-year-old.
A photo of the twins - a male and a female - has received over a thousand congratulatory comments on Facebook.
The cubs are currently "very fragile and need time to stabilise" under 24-hour intensive care, according to their caretakers.
"We are all looking forward to meet the giant panda cubs. Please wait a few months patiently to make their debut and officially meet everyone!" Ocean Park Hong Kong said.
Of Ying Ying's twins, the female cub appears to be more fragile because her body temperature is lower and her cries are weaker compared to her brother, the park operator said.
She weighs just 122g.
Ying Ying was "understandably nervous" during birth and spent most of the time lying and twisting on the ground, Ocean Park said.
Giant pandas are notoriously reluctant to mate. Ying Ying and the twins' father, Le Le, have been housed at Ocean Park since 2007, when they were gifted to Hong Kong by Beijing. They mated successfully in March.
China's long-term conservation effort has reversed the population decline of giant pandas. They are now considered a vulnerable species and no longer endangered.
Aside from serving as theme park attractions, pandas have also figured in China's diplomatic efforts.
Joel Guinto