Earlier, some Austrians let out welcoming cheers as busloads of refugees pulled up on their border with Hungary -- and weary passengers clutching children streamed toward them.
The passengers carried their meager belongings in backpacks as they exited the vehicles in the rain.
The Austrian Red Cross also provided medical supplies and warm blankets.
At least 5,500 refugees have arrived in Austria from Hungary since Friday night, the United Nations refugee agency said Saturday. Some 2,500 are still in the border town of Nickelsdorf and are waiting to head to the capital, Vienna, the UNHCR said via Twitter.
Deputy Chief of Burgenland State Police Werner Fasching earlier said about 10,000 migrants were expected in total. There are only enough beds for 600 people in and around Nickelsdorf, and the bulk of the refugees are being sent to Vienna via trains and buses, he said.
"We are trying to move as many as possible in the direction of Vienna," Fasching said. There the migrants will receive food, drink and, if needed, medical care. Some who wish to continue on to Germany will be permitted to do so.
Their arrival in Austria caps an emotional week for the migrants, many of whom had walked for hours before they got into dozens of buses provided by the Hungarian authorities.
But the busing was only a temporary solution for this band of refuge seekers, leaving questions about what will be done for the thousands of other Syrian and other migrants still crossing the Mediterranean and traveling north through Europe.
In light of this week's acute situation, Austrian and German officials agreed to allow thousands of migrants into their countries, Austrian Chancellor Werner Faymann said.
The UNHCR said it "welcomes the decision of Austria and Germany to receive thousands of refugees and migrants who crossed the border last night from Hungary. This is political leadership based on humanitarian values."
And it's not just the politicians who have extended a hand. Some individuals in Vienna are donating train tickets for refugees heading onwards across Europe, the UNHCR said, while others elsewhere are giving food and supplies.
UNHCR spokeswoman Melissa Fleming said the refugees she'd met at a Vienna station were enormously grateful for the welcome they'd received since reaching Austria.
Chaotic scenes erupted Thursday as trains packed with Syrian refugees hoping to travel to Austria or ultimately Germany were abruptly halted at Bicske station outside the Hungarian capital of Budapest.
Hungarian authorities wanted to send them to a nearby holding camp, but -- fearing that once there they would be badly treated and unable to continue their journey north -- the migrants refused to get off the train despite suffocating heat and limited food.
After a standoff lasting more than a day, hundreds set off on foot Friday along the train tracks toward the Austrian border, about 100 miles away. Hungary eventually sent buses to pick them.
About 300 more at Bicske station agreed to go to the nearby refugee camp, according to a Hungarian government statement.
Amid the chaos, the U.N. refugee agency said a 50-year-old man had died Friday in Bicske. Initial reports suggest he fell and hit his head on the tracks while trying to run away from police, said Montserrat Feixas Vihe, the UNHCR's Central Europe representative.
More than 1,000 other refugees set off on foot from Budapest's main Keleti station -- where they had been waiting for days to travel onward -- and walked for hours beside a highway. They also were picked up by buses, as were others still at the station.
Over the past week, Keleti station had become a focal point of the crisis engulfing parts of Europe as an unprecedented wave of people -- mostly refugees fleeing conflict in Syria, Iraq and Afghanistan -- seek to reach Northern and Western Europe.
The refugees on the buses were exhausted but relieved to be on their way, though some expressed doubt that they would actually be taken as far as Austria. "Are they taking us to border?" one man asked suspiciously.
Once in Austria, they plan to head to various parts of Europe -- Germany, Italy, Sweden were among the destinations they named.
Christian Stella, a police commander at the Austrian border, said those refugees who request asylum in Austria are being taken to the Nova Rock stadium in Nickelsdorf.
Those who want to go on to Germany are either being put on special trains to Vienna from Nickelsdorf train station, or bussed directly to Vienna from the border crossing, he said. About 2,500 people had been processed by around noon.
Austria's Interior Ministry warned on its website that it is illegal for private individuals crossing into Hungary to pick up migrants and ferry them back to Austria.
The ministry has, however, asked for Austrian citizens and companies to lend the government land on which to put temporary accommodation for refugees.
Hungary PM: 'If we let everyone in, we will destroy Europe'
Hungary's right-wing government is erecting a barbed-wire fence along its more than 100-mile long border with Serbia in a bid to prevent migrants crossing illegally.
And on Friday, lawmakers passed a package of bills -- due to take effect September 15 -- aimed at tightening border restrictions to prevent migrants from entering illegally, Hungarian news agency MTI reported.
Some lawmakers wanted an amendment that would have allowed army units to back up border police. That was defeated Friday, but legislators intend to vote on the measure again later this month, MTI reported.
Hungarian Prime Minister Viktor Orban said Hungary's borders -- and Europe's -- must be controlled, or else "tens of millions" will enter.
"The supply of immigrants is inexhaustible," Orban said Saturday. "If we let everyone in, we will destory Europe."
Earlier this week, Orban said that without strict border controls, EU migrant quotas are "an invitation" for migrants to come. "Turkey is a safe country, why don't you stay there?" he said, adding that migrants who reach Serbia should also stay put.
Despite Hungary's hardline, former Prime Minister Ferenc Gyurcsány told CNN's Christiane Amanpour that his family has allowed about a dozen or so migrants to stay at his home after realizing that giving them food was not enough. He said he has cooked meals of chicken and vegetables for families with babies. His family has washed the clothes of migrants and taken them to doctors.
"I do what I can do," Gyurcsány said.
The migrants don't know who he is, Gyurcsány said.
"It's not important in this case," he said. "This is my private story. It is provided by my family."
Despite finding itself on the front lines of the migrant crisis, Hungary is more a transit point than a destination on a long journey to wealthier nations such as Austria and Germany, where the refugees hope to claim asylum.
Speaking to reporters before a meeting of EU foreign ministers Saturday in Luxembourg, Hungarian Foreign Minister Peter Szijjarto defended his country's response.
"What has been happening in Hungary are two things -- first, the failed migration policy of the EU, and the second one is series of some irresponsible statements made by European politicians," he said.
Szijjarto also accused the migrants of exacerbating the situation by failing to cooperate with authorities or go to refugee camps where they could get shelter and other essentials. In setting off on foot along the country's main railway line and highway, they then triggered an emergency situation, he said.
"That is why we have decided to send buses and deliver them to the Austrian border where they wanted to go."
Under European law, those seeking asylum are approved in the country where they first registered, and most migrants prefer to file paperwork in Western European nations, which have better programs set up to help refugees.
Still, Hungary has been inundated with 140,000 asylum applications since January, and there have been about 2,000 new arrivals daily, U.N. refugee agency spokesman Babar Baloch said.
Hungarian authorities have said under EU legislation, they can't allow people to travel internationally without the proper documentation -- a valid passport, a ticket and any necessary visas.
Mogherini: Legal and moral duties
Addressing reporters after Saturday's Luxembourg meeting, EU foreign policy chief Federica Mogherini said Europe is finally starting to see the problem for what it is -- not just affecting certain EU states, such as Italy and Greece, but a Europe-wide issue.
"We also have to start using the right words. It is partially a migrant flow, but it is mainly a refugee flow, which puts us in a different situation when it comes to our legal and moral duties," she said. It's also a situation that is "here to stay," she warned.
"The time for blame games is over," Mogherini said, and EU states need to find common ways of sharing responsibility for the influx of migrants and refugees, rather than leaving only a few to shoulder the burden.
Mandatory quotas have been rejected by some EU members. However, a voluntary system makes decisions more difficult, she said, so common systems are needed to speed up the process.
More must also be done to tackle the networks of people traffickers exploiting migrants who seek to cross the Mediterranean from North Africa to Italy, or to travel from Greece via the Balkans to northern Europe, she said. Restoring stability to Libya and finding a political solution to the war in Syria are key to resolving the issue longer term, she said.
Europe must remember its own history of wars and conflicts, she added, as it considers its responsibility to help and protect refugees.
'The problem is a German problem'
Orban, the Hungarian Prime Minister, said earlier this week that the situation was not of his country's making.
"The problem is not a European problem; the problem is a German problem," he said.
Germany's government said last month it expected up to 800,000 asylum seekers to come this year -- four times more than in 2014. But Orban said German Chancellor Angela Merkel had said that they must be registered before leaving Hungary, which his country was trying to do.
Orban said his country was just trying to enforce EU rules.
"Don't criticize Hungary for what is being done. Let Hungary do the job as it is written in the European regulations," he said.
While European leaders struggle to come up with a coherent plan, the men, women and children caught up in the crisis continue to suffer.