Around 50,000 people arrived in Greece in July alone, the organisation says.
Greece's leader said the country was unable to cope, and called for EU help.
Separately, Italian police arrested five suspected traffickers over the deaths of about 200 people after a migrant boat sank on Wednesday.
They included two Libyans, two Algerians and a Tunisian, held on suspicion of multiple murder and people trafficking.
Survivors have said that traffickers used knives to slash the heads of African migrants and belts to thrash Arabs to keep them in the hull.
In other developments:
- Some 380 people rescued from a fishing boat were brought to Sicily, a day after being rescued
- The UNHCR strongly criticised the British and French governments over the situation in Calais, where 3,000 migrants are living in makeshift camps, saying it should be treated as a "civil emergency"
- A Sudanese national has been charged after allegedly walking nearly the full 50.5 km (31-mile) length of the Channel Tunnel towards the UK
- Austrian
authorities have stopped taking any more migrants at the country's main
reception camp in Traiskirchen
The organisation'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, he said, but Greece must "lead and co-ordinate".
"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".
Syrian Mustafa Mohammad and
this young boy came ashore on Lesbos on Thursday "The risk to a child forced to sleep on the street of being abused, or of a baby dying of heatstroke, is very real," said Kitty Arie from the charity.
"This is Europe in 2015. We can't leave these children in this desperate situation."
Lesbos is seeing an influx of
Syrians and Afghans like these people helped ashore from a dinghy
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 neighbouring Hungary - both EU member states and both in the border-free Schengen zone, where passports are generally not checked.
EU border agency Frontex says it has not received enough pledges of assets
from EU states to help Greece and Hungary with the current influx of migrants. Meanwhile, the EU is struggling to cope with the thousands of migrants heading for Italy aboard unseaworthy vessels from Libya, where people traffickers are charging huge sums to smuggle them into Europe.
On Thursday, Irish navy ship LÉ Niamh docked in Sicily with 367 migrants - mostly Syrians - who were rescued from Wednesday's sinking.
