The Austrian chancellor has said his country and Germany have agreed to let migrants cross their borders from Hungary.
They include hundreds who defied attempts to register them in Budapest and began walking to the border.
European Union states are struggling to agree on how to deal with an unprecedented surge in migrants.
Budapest's surprise decision to lay on buses followed several days of chaotic scenes.
Those travelling on foot were among thousands camped outside Keleti station in Budapest, while another group fled along railway tracks in Bicske, to the west of the Hungarian capital, from a train stopped by police.
The BBC's Matthew Price, who has been travelling with those walking along the main road to Vienna, said there was fear among the migrants that they would be arrested rather than sent to the border.
Explaining the move, Hungarian Prime Minister Viktor Orban's chief of staff, Janos Lazar, said the government was despatching buses so the country's transport network would not become "paralysed".
Buses have also been arriving to collect migrants who have been camped outside Keleti station in central Budapest.
They are to take people to Hegyeshalom, on the border with Austria.
Mr Faymann said he would allow the migrants continued passage because of the "emergency situation" on Hungary's borders.
He said the decision came after talks with his German counterpart, Angela Merkel, adding he expected Hungary to respect any EU quotas for asylum seekers.
Hungary, along with the Czech Republic, Hungary, Poland and Slovakia, issued a joint statement on Friday opposing a quota system.
The Red Cross has said it expects to receive 800 to 1,500 people to arrive at one of its migrant centres in the early hours of Saturday.
An Austrian interior ministry statement (in German) said they would have the opportunity to apply for asylum in Austria, the statement said.
Hungary has become a major transit nation for people fleeing the Middle East and Africa, seeking to reach north and west Europe.
In other developments:
- The body of three-year-old Alan Kurdi, images of whose body sparked global outrage, was buried in Kobane, Syria
- There were clashes at Keleti station after far-right extremists threw two firecrackers towards migrants, sparking an angry response
- Hundreds of people broke out of a refugee camp at Roszke, near the Serbian border, and were being pursued by police. Video from the camp showed clashes between migrants trying to break out and riot police, who used spray
- Hungarian MPs approved tougher border controls and penalties for migrants trying to pass through to Germany
- John Tory, the mayor of the Canadian city of Toronto, said that he would personally sponsor an asylum application for a Syrian family.