She said Germany would speed up asylum procedures and build extra housing, but called on other EU states to help.
French President Francois Hollande said quotas for EU countries to relocate 120,000 migrants were being planned and that France would take 24,000.
Meanwhile, the flow of migrants across Europe shows no sign of easing.
On Monday, large numbers of people were reported to be streaming into Hungary across its southern border with Serbia.
The 160,000 are said to include 66,000 who have arrived in Greece, 54,000 in Hungary and 40,000 in Italy.
In other developments:
Greece has requested emergency EU assistance to deal with migrants arriving from Turkey
Spanish media say police fired rubber bullets at migrants in a detention centre in Valencia after about 50 tried to escape
Police in Macedonia scuffled with thousands of migrants trying to cross into the country from Greece
Migrants sit in the road in front of the police line as photographers mingle with them.
An empty coach waits to pick them up and start the journey to Budapest. There are about 400 people here.
A voice on a loudspeaker tells the migrants in English that buses will come soon.
They are not convinced. They continue their sit-down protest but police warn them that buses can’t get through if they don’t move.
Other exhausted families trudge along the railway line, heading to the border camp.
Thousands of migrants who had arrived in Hungary made their way through Austria to Germany over the weekend. Those arriving at Munich station were cheered by locals.
Mrs Merkel thanked volunteers who had helped and welcomed those arriving, saying they had “painted a picture of Germany which can make us proud of our country”.
However, she said that although Germany was “a country willing to take people in”, it was “time for the European Union to pull its weight”.
Germany – which expects 800,000 asylum requests this year – could face costs of €10bn (£7.3bn) next year because of the influx, she added.
New quotas drawn up by the European Commission are set to be unveiled on Wednesday.
Hungary had previously blocked migrants travelling to Western Europe, but dropped restrictions on Friday after struggling to cope with thousands camping in its capital, Budapest.
It is continuing work on a fence along its border with Serbia and its parliament passed tough new legislation on illegal immigrants last week.
Speaking on Monday, Hungarian Prime Minister Viktor Orban said “as long as we can’t defend Europe’s outer borders, it is not worth talking about how many people we can take in”.
Those migrants trying to reach Germany were seeking a “German life” rather than physical safety, he said, adding that if the stream continued it would endanger Europe’s “Christian welfare states”.
Mrs Merkel has become a hero to many migrants for allowing large numbers to cross into the country from Hungary. But the Bavarian Christian Social Union, a sister party to Mrs Merkel’s Christian Democrats, have accused the chancellor of sending a “totally wrong signal”.
On Sunday night, there were two fires at accommodation centres for asylum seekers in Germany, with police confirming that one was “politically motivated arson”.