It happened inside an enclosure during a supervised visit on Thursday.
The animal mauled the girl and her mother as she went to save her. They were given first aid at the scene before being taken to Temple Street Children's Hospital and the Mater Hospital.
It is understood the youngster has undergone surgery for injuries to her arm and abdomen.
A statement from Dublin Zoo described it as "an unfortunate accident", adding: "The Dublin Zoo first aid team was at the scene immediately and attended to the visitors.
"The welfare of our visitors and animals is of paramount importance and all steps are taken by our experienced staff to ensure optimum safety.
"Dublin Zoo would like to underline this was very much an isolated incident. The staff of Dublin Zoo are deeply upset by this incident and we reiterate strongly that their utmost concern lies with the visitors and their speedy recovery."
Tapirs are of a similar size to pigs and look like anteaters.
The zoo said an investigation into the incident is underway.