A year ago I compared a photograph form the 1904 St. Louis World's Fair to a 1914 postcard of the construction of the St. Louis City Public Library headquarters.
According to this map of the fairgrounds, the Sunken Gardens were in Forest Park. (Most of the fair was, and I should have recalled this fact.) If you enlarge the image, you can see it in the center-left, by the northwest corner of the lagoon. Between the Liberal Arts building and the Mines and Metallurgy building. (D2)
The library has a new gallery of images from their construction, including this photo taken in 1908, indicating that their Sunken Gardens were created at the same time the library was built.