Around 3,000 people joined a 10-kilometre (6.2-mile) run which was part of a series of programmes marking the birthday.
Visitors, many of them children with parents, cheered Yuan Zai when she was presented with a birthday cake -- made of apples, pineapples, carrots and buns and prepared by the zookeepers.
The main attraction was when the cub grabbed different cards in "Zhua Zhou", a traditional crawling game for one-year-old babies in many Chinese communities.
Yuan Zai initially picked up the card for painter, among a variety of others.
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The cub made her public debut in January and since then the exhibition centre at Taipei Zoo has often been swamped with fans.
In the six months to June 2.4 million people visited the zoo, about a 50 percent rise over the same period of 2013.
Yuan Zai was delivered on July 6 last year following a series of artificial insemination sessions because her parents -- Tuan Tuan and Yuan Yuan -- failed to conceive naturally.
Mother and daughter were reunited for the first time on August 13, a meeting that saw the giant panda licking and cuddling her baby before they fell asleep together inside a cage.
Tuan Tuan and Yuan Yuan, whose names mean "reunion" in Chinese, were given to Taiwan by China in December 2008 and have become star attractions at Taipei Zoo, as well as a symbol of warming ties between the former bitter rivals.
Fewer than 1,600 pandas remain in the wild, mainly in China's Sichuan province, with a further 300 in captivity around the world.