Five-year-old Sumatran tiger Melati gave birth to the cub on 22 September after a six-minute labour.
The pregnancy, which lasted about 15 weeks, was kept secret by zookeepers who are "over the moon" about the cub.
It was born six months after the opening of the "tiger territory", designed to encourage the endangered sub-species of tiger to breed.
BBC correspondent Emma North said the cub was the product of a "whirlwind romance".
Senior zookeeper Paul Kybett explained: "They came from opposite ends of the globe. Melati joined us from Perth at the end of September last year.
"Jae Jae came from Ohio. When they first saw each other they made the lovely sneezy purr kind of noise that they make.
"When we introduced them for the first time he trotted up to her and gave her a big head rub so we knew we were going to be getting something quite special."
He added: "When it came to her due date, we were all watching our monitors with bated breath.
"The actual birth happened very quickly and Melati's maternal instincts kicked in immediately.
"We couldn't have asked for a smoother birth.
"It's still very early days, so we're leaving Melati to take care of her adorable baby, and our cameras allow us to watch them both from a distance.
"So far she's proving to be a doting mum."
Malcolm Fitzpatrick, curator of mammals at the zoo, said the birth was very important for the global zoo tiger population.
He said: "This tiger's one of the most important births in Europe this year."
Zookeepers do not yet know the sex of the cub, which is in the exhibit's cubbing den with its mother and will remain away from the public for several weeks.
The cub is the grandchild of the zoo's last tiger cub, Hari, the father of Melati.
Hidden cameras have captured the moment London Zoo's first tiger cub for 17 years was born.