It was uplifted to make room for a new Tasmanian Devil exhibition in 2013, which created an uproar.
The following year, then-Mayor Len Brown apologised to the people and Mayor of Fukuoka City, and to Auckland's Japanese community at the Japan Day event in Auckland.
The restored garden was opened in July at its new location in Western Springs Lakeside Park, replacing the original one built in Auckland Zoo.
A Friends of Fukuoka Garden group spokesman said the vandalism was "heartbreaking".
"It is particularly disappointing considering that we had spent four years keeping the lamp safe from any damage for the restored garden," he said.
"The Japanese community living in Auckland are used to seeing things and places being vandalised, but this is still heartbreaking."
Features of the original garden that had been incorporated into the new garden include four bonsai plants, paving stones, lanterns, a water basin and the entrance feature.
The new garden was 2.6 times larger than the original, and also has a tea pavilion, waterfall and pond.
The spokesman said the group also wanted more security around the garden.
"What happened shows the council needs to increase security," he said.
"There needs to be more LED lighting, CCTV and lockable gates around the garden area."
McCormack said the council was working with its insurers and the group to have the damage repaired as soon as possible.
"We are looking at further security options in the wake of this saddening event," she added.
"The costs involved have yet to be calculated."