Meanwhile, Bavarian media has reported that police in Munich are preparing for the arrival of some 10,000 refugees.
The pre-dawn move eased the immediate pressure on Hungary, which has struggled to manage the flow of thousands of migrants arriving daily from non-EU member Serbia.
But officials warned that the numbers arriving in Hungary from the south was still rising, as more westward-bound travelers continued to arrive in Budapest within hours of the mass evacuation of the capital's railway station on Friday.
Many of the asylum-seekers had already spent several months in refugee camps in Turkey, and had taken long journeys by boat, train and foot through Greece and the Balkans. Some had to crawl under barbed wire on Hungary's southern frontier.
Hungary refused to grant passage to the refugees throughout last week, painting the ongoing crisis as a defense of Europe's prosperity, identity and "Christian values" against an influx of Muslims.
Austria and Germany had announced on Friday night that they would take full responsibility for the refugees that were already on the move west or had camped out in their thousands at Keleti train station, marking a breakthrough in the refugee crisis.
"We're happy. We'll go to Germany," said a Syrian man who only gave his name as Mohammed.
Another man, who declined to be named, told Reuters news agency: "Hungary should be fired from the European Union. Such bad treatment."
Hungary caves in after week-long stand-off
Hungary was forced to relent after demanding that travelers report to government-run asylum centers after being challenged by defiant migrants coming largely from Syria, Iraq and Afghanistan.
Under European Union accords, refugees are to report asylum at their first point-of-entry and are expected to have their asylum claims processed there.
The stand-off over refugees led Hungary to suspend its train services to Austria and Germany from Budapest's Keleti station on Tuesday, compounding the build-up of migrants there. Thousands of migrants stranded at the station then decided to march west on Friday along Hungary's major motorway. Hundreds more broke through police lines in the western town of Bicske, where police were trying to take them to a refugee camp.
Janos Lazar, chief of staff to Hungary's prime minister, justified the move by saying that the migrants' movements across the country on Friday had been putting rail services at risk while also causing massive traffic jams. Hungarian officials continue to call for a quick solution in the ongoing refugee crisis.
More than 140,000 migrants have been recorded crossing into Hungary so far. Determined to stem the tide, Hungary has built a 3.5-meter (11.5-foot) high fence along its border with Serbia. On Friday, the Budapest parliament adopted measures the government says will effectively seal the frontier to migrants as of September 15.