According to the agency, the refugees are forced to leave their homes due to inadequate accommodation, water and sanitation, the report says.
Around 50,000 people arrived in Greece in July alone, the organization says.
Greece's leader said the country was unable to cope, and called for EU help.
The UNHCR says nearly all new arrivals in Greece are refugees from the wars in Syria, Iraq and Afghanistan.
The organization’s European Director, Vincent Cochetel, said facilities for the refugees on the Greek islands were "totally inadequate", after more migrants arrived in Greece in July 2015 than in the whole of the previous year.
"Greece's EU partners must do more to ease the burden, but Greece must "lead and co-ordinate," he said.
"On most of the islands there is no reception capacity; people are not sleeping under any form of roof. So it's total chaos on the islands."
"After a couple of days they are transferred to Athens, there is nothing waiting for them in Athens," he complained.
Greek Prime Minister Alexis Tsipras said the problem "surpasses" Greece's abilities, and that his country's economic problems meant it was facing a humanitarian "crisis within a crisis."
Separately, another UNHCR official described conditions at Austria's main reception camp in Traiskirchen, 20km (12 miles) south of Vienna, as "intolerable, dangerous and inhumane."
About 4,500 people are at the camp, which was built to house 1,800. Many are now sleeping in the open.
Most arrived in Austria via neighboring Hungary - both EU member states and both in the border-free Schengen zone, where passports are generally not checked.
