The influx has piled pressure on Greece's services at a time when its own citizens are struggling with harsh cuts and its government is negotiating with the EU and the IMF for fresh loans to stave off economic collapse.
Boatloads of migrants arriving every day had triggered a "humanitarian crisis within the economic crisis," Tsipras said after a meeting with ministers.
"The EU is being tested on the issue of Greece. It has responded negatively on the economic front - that's my view. I hope it will respond positively on the humanitarian front," he said.
The comments came as the United Nations refugee agency (UNHCR) called on Greece to take control of the "total chaos" on Mediterranean islands, where thousands of migrants have landed.
About 124,000 have arrived this year by sea, many via Turkey, according to Vincent Cochetel, UNHCR director for Europe.
"The level of suffering we have seen on the islands is unbearable. People arrive thinking they are in the European Union. What we have seen was not anything acceptable in terms of standards of treatment," Cochetel said after visiting the Greek islands of Lesbos, Kos and Chios.
"I have never seen a situation like that. This is the European Union and this is totally shameful," he added.
Squalid conditions
At a makeshift refugee centre at Kara Tepe, a hilltop about 5 km north of Lesbos island's main town of Mytilene, about 50 white tents provided by the local council struggled to accommodate the waves of people coming in daily.
Rubbish littered the area and locals said 16 toilets were frequently blocked despite attempts by authorities to keep the area clean.
Up to 10 people could be seen sharing one of the tents, while others lay on pieces of cardboard, jostling for space under the shade of olive trees in sweltering heat.
"The government had battles on plenty of fronts and probably could not give as much attention to the problem," the island's mayor Spiros Galinos told Reuters.
The UNHCR's Cochetel said Greece had to step up its response.
"We've told the Greek authorities that if it was a natural disaster, there would be mobilisation of other assets including from the ministry of defence," he said.
"It's easy, there are plenty of empty army barracks in Greece, there is plenty of uncultivated land that could be rented and sites could be developed."
Tsipras said the problem was a European issue and not just a Greek one. He said ministries would coordinate to help relieve the burden on distant islands and that a special unit would be set up to make use of dedicated EU funds to beef up border controls and integrate migrants into society.
"The immigrant flow to Greece is beyond of what our state infrastructure can handle," Tsipras said. "We have significant problems to face it and that's why we have asked help from EU."
The country, stuck in a seven-year financial crisis, has returned to recession again and narrowly averted bankruptcy this year by agreeing a bailout deal.
The European Union has sought to share the burden of the refugees across its member countries, but the response has been mixed. EU leaders have pledged to relocate 16,000 migrants over two years, which Cochetel called "far too little and too late".
Britain has said it will not participate. It is currently struggling with its own crisis as thousands of migrants seek to enter via the Channel Tunnel.
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