Sergeant Hugh Tait, who was controlling the scene, said the crowd, mainly students, complied with the police request to climb down, and were "moved on", about 10.40pm.
One university student was arrested on a charge of disorderly behaviour following a "big party" celebrating St Patrick's Day at the three flats near the intersection with Cumberland St.
"They were pretty good kids - it just got a bit out of control," Sgt Tait said.
"It had potential to go horribly wrong."
A bystander said police entered the flats to communicate with revellers, which helped keep the situation calm.
He said some students had been climbing between the veranda and the three balconies above.
Senior Sergeant Bruce Ross said a member of the public called police to say that between 50 and 100 students were on the veranda.
The person feared the weight of the students could cause it to collapse.
A 19-year-old second-year commerce student told the Otago Daily Times he was on the veranda because "everyone was up there" and it "looked cool".
Asked about the safety concerns, he said: "If I got hurt, everyone would."
He said he could see the structure was sagging, but was not worried.
When the ODT left the scene about 10.50pm, students were being moved along Dundas St, towards Clyde St, chanting, although the scene was good natured.
