Yuan Zai lumbered up and down the wooden structure inside an exhibition enclosure while mother Yuan Yuan was munching bamboo.
Yuan Zai entertained the crowds for 40 minutes before falling asleep.
"Her muscles are getting stronger and stronger. It is no problem for her to crawl up and down the structure," Taipei Zoo spokesman Chao Ming-chieh said.
"But whenever her activity slows down, then she is telling you that she needs a snap."
The exhibition centre at the Taipei Zoo was swamped with fans -- many of them parents with their children -- keen to get the first pictures of the cub.
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She weighed 180 grams at birth, but now weighs around 14 kilograms.
Zoo authorities said 1,200 visitors flocked to the enclosure within minutes of the 9:00 am opening time.
Each visitor was permitted to stay around the enclosure for a maximum of 10 minutes, with zoo keepers limiting the total number entering each day to 19,200.
Zookeepers had to separate tiny Yuan Zai from her mother a few days after birth. They raised her in an incubator with round-the-clock monitoring after she was slightly injured in the leg.
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.