Chancellor Angela Merkel, speaking after a weekend in which 20,000 refugees entered Germany from Hungary by train, bus and on foot, described the scope of the migration as "breathtaking" and tried to reassure German citizens that the crisis was manageable.
"I am happy that Germany has become a country that many people outside of Germany now associate with hope," she said at a news conference in Berlin.
But she and her vice chancellor, Sigmar Gabriel, coupled their message of optimism with a warning to European Union partners who have resisted a push from Berlin, Paris and Brussels to agree quotas for refugees flowing in mainly from Syria, Iraq and Afghanistan.
"What isn't acceptable in my view is that some people are saying this has nothing to do with them," Merkel said. "This won't work in the long run. There will be consequences although we don't want that."
Gabriel said that if countries in eastern Europe and elsewhere continued to resist accepting their fair share of refugees, the bloc's open border regime, known as Schengen, would be at risk.
"This would be a dramatic political blow for Europe, but also a heavy economic blow, also for those countries that are saying they don't want to help now," he said.
At Roszke, on Hungary's border with Serbia, around 300 refugees broke through a cordon around a reception camp and set off down the wrong side of the motorway towards the capital Budapest, Reuters witnesses said.
Police were unable to prevent their escape despite using pepper spray as refugees scuffled with officers.
Only months after Europe narrowly averted a Greek exit from the euro zone, the refugee crisis has emerged as the bloc's biggest challenge.
European Commission President Jean-Claude Juncker is due to unveil new proposals on Wednesday on how to distribute refugees among member states.
An EU source told Reuters that under his plan, Germany would take on more than 40,000 and France 30,000 of the 160,000 asylum seekers the Commission says need to be relocated from Italy, Greece and Hungary, the main entry points to the EU for refugees arriving by sea and land.
The 160,000 that Juncker wants to redistribute within the EU are just a fraction of the hundreds of thousands of refugees from Asia, Africa and the Middle East who have reached Europe this year on leaky boats across the Mediterranean or over land through the Balkan peninsula.
Germany has announced it is letting Syrians seek asylum regardless of where they enter the EU, suspending normal rules and accelerating the movement of refugees north and west from the edges of the bloc.
Just last month, more than 100,000 asylum seekers reached Germany, which is preparing for 800,000 this year, around one percent of its population.
France said it would take in up to 1,000 refugees currently in Germany as a matter of urgency, before the adoption of the European Commission plan.
Britain would resettle 20,000 refugees from Syria over the next five years, Prime Minister David Cameron announced, responding to a growing public clamor for his government to do more to help.
Other countries have resisted taking in refugees, with Poland arguing that it is already burdened by people from Ukraine, and Slovakia saying it preferred to accept Christians.
Hungary's Prime Minister Viktor Orban, a right-wing populist whose hard line during the crisis has drawn both praise and criticism, reiterated his opposition to quotas on Monday, calling this debate premature. Juncker's proposal would exempt the main entry countries — Hungary, Greece, and Italy — from taking relocated refugees.
Hungary's defense minister, Hende Csaba, resigned because the armed forces were being too slow in building a border fence to keep out refugees.