"This has to be an eye-opener how messed up the situation in Europe is now," Kurz said as he arrived in Luxembourg for informal talks with his EU peers dominated by the deepening refugee crisis.
"I hope that this serves as a wake up call that (the situation) cannot continue."
Thousands of exhausted migrants were streaming into Austria from Hungary early Saturday.
Several thousand migrants had been blocked for several days in the main railway station in Budapest, the Hungarian capital, as the government insisted they could not travel onwards without proper documentation.
Conditions at the station became increasingly tense, with more than a 1,000 migrants setting off on foot for Austria late Friday in defiance of the authorities.
Austria expects more than 10,000 refugees to arrive on Saturday.
"Thank god, the problem could be solved yesterday evening in a humanitarian way," Kurz said, warning that this was only a one-off solution to an immediate problem.
"Anyone who believes that you can sit out this problem is wrong," he said.
The 28-nation EU is sharply divided over what to do with the flood of migrants fleeing war and turmoil across the Middle East and North Africa.
Germany has led efforts to open the doors, saying it would accept 800,000 refugees this year and backing plans for mandatory quotas in the EU.
Hungary, along with many of the bloc's newer eastern members, flatly opposes quotas and insists current rules should be applied whereby asylum seekers must be processed in the country they first arrive in, not the country they want to go to.
In this case, most of the refugees are Syrians arriving via Greece which has been overwhelmed by the numbers.
Hungarian Foreign Minister Peter Szijjarto said the problems in Hungary had been caused by "the failed migration policy of the EU and ... irresponsible statements made by some European politicians."
Szijjarto did not elaborate but Hungarian Prime Minister Viktor Orban bluntly blamed Germany earlier this week for encouraging people to risk their lives coming to Europe with its promise of more places for refugees.