Speaking to Britain's Observer newspaper, Kurz said it was “unacceptable in the long run” that Greece is waving refugees through to Europe.
"The European Union cannot act like a human trafficker,” he said.
He went on to describe the situation at the Macedonian and Greek border, where 30,000 refugees and migrants trying to head to northern Europe are currently stuck, as ‘absurd'.
He added: “You have people protesting and in some cases resorting to violence because they want to move from an EU state into a non-EU state. It shows what grave mistakes some European politicians have made over the last 12 months. Opening the borders in 2015 has led to an influx that Europe can no longer shoulder.”
Also appearing Sunday night on Germany's TV channel Das Erste, on the Anne Will programme, the 29-year-old minister spoke of the options to be discussed at the meeting of the European Council with Turkey taking place today in Brussels.
"We have two options: either we leave our borders uncontrolled and open - and more migrants/refugees will follow suit - or we stop them at our borders,” he said.
“There is morally no higher value case to be made when stopping refugees/migrants in Turkey or in Greece.”
“The closure of Balkan route will be agreed upon at the European Council meeting on March 7th. There will be an end to the policy of waving through people through Central Europe."
He also added that the line between those who seek protection and those seeking a better life has become blurred.
"It is paramount to protect those in need for protection, but the right to protection does not entail the right to choose the country of protection," he said.
A further 100,000 refugees in Greece by the end of March
His comments follow warnings from Europe's migrant commissioner that Greece could soon see a further 100,000 refugees and migrants at its door.
"Hundreds are arriving on a daily basis and Greece is expected to receive another 100,000 by the end of the month," Dimitris Avramopoulos told a conference on Saturday in the Greek capital Athens.
Austrian volunteers supporting refugees in Greece have also written an open letter to the Austrian government, saying their current policy of closing borders and encouraging Greece to register and retain new arrivals could trigger a “humanitarian disaster”.
They argue in the letter that measures to try and prevent people from crossing borders into and within Europe will not “have any effect, unless the acts of war are put to an end”.
“Otherwise, traumatised, terrorised people will continue to do anything to escape their misery,” they write.
Border controls could cost €210 million
Meanwhile, research from Munich's Ifo-Instituts suggests that Austria's border policy, which has seen the introduction of stronger controls along the country's southern borders, could cost as much as €210 million, the equivalent of paying the basic care for 10,000 asylum seekers in Austria.
The costs of allowing unregulated migration, however, could well be much higher and ÖVP Finance Minister Reinhold Mitterlehner said the question of cost should be reversed.
“If too many refugees flow into the country unrestrained and uncontrolled, that would cost our system more than border controls,” he said.