Police say they were called to the scene shortly after 7.30am.
It finally came to a stop after hitting the covered walkway in the lower part of the Octagon, leaving woodchips and debris in its wake.
At least nine police were at the scene, along with 2 fire trucks and an ambulance.One person was taken to hospital with moderate injuries.
Police said the driver had possibly suffered a medical event.
A council spokesman said staff and contractors were on site, working to make the area safe.
"There is damage including to the pedestrian canopy, fences and also some damage to at least one of the Octagon plane trees," the spokesman said.
"The focus for now though is on making the site safe. Structural and tree health assessments will follow once it is safe to do so.
A Fire and Emergency NZ spokeswoman said firefighters from the Dunedin and Willowbank stations were sent to the scene about 7.30am.
The firefighters and ambulance services helped one person still in the truck to get out, she said.
About midday Saturday, contractors were still sweeping up woodchips with shovels or spades.
An excavator was also being used in the clean-up.
The scene continued to attract curious observers.
The truck was towed from the scene at about 1.45pm.
The road through the Octagon reopened at about 4.30pm after being closed to traffic for the day, but the footpath on the lower side remains fenced off to pedestrians.