Mrs Merkel`s Christian Democrats and junior coalition partners the Social Democrats have quarrelled for weeks over the issue.
Germany says it expects to receive at least 800,000 asylum seekers this year.
Earlier, the European Commission said that three million migrants were likely to arrive in Europe by the end of 2017.
The huge influx of asylum seekers has caused political turmoil across the EU with member states disagreeing about how to deal with the crisis.
"We took a good and important step forward," said Mrs Merkel, whose open-door refugee policy has come in for strong criticism in Germany.
The five special centres would hold migrants from countries deemed safe; those barred from re-entering Germany; and those refusing to co-operate.
In an accelerated asylum process, cases could be heard in a week, not months, and appeals would take only a further two weeks. Most could expect to be deported.
At a news conference, Mrs Merkel and Social Democrat leader Sigmar Gabriel also stressed the importance of tackling the reasons causing people to flee their countries and of securing the EU`s external borders.
The UN says the onset of bad weather has failed to stem the flow of migrants risking perilous sea journeys to Europe.
Peter Sutherland, UN Secretary-General Ban Ki-moon`s special representative on migration, told the BBC that the Syrian war was "driving people to desperation in terms of leaving and it will continue in its effects".
Conflicts and abuses in Iraq, Afghanistan, Eritrea and Somalia are also driving people towards Europe.
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