BERLIN—Germany started imposing document checks for people entering the country Sunday, seeking to stem the tide of migrants streaming in by suspending a cornerstone of European integration: open borders.

The temporary border controls—announced on the eve of a European Union interior ministers’ meeting on Monday to find measures to cope with the escalating migration crisis—came as a warning shot to other countries in the bloc. It indicated that Germany, the EU’s largest country and top destination for refugees and other migrants, was prepared to undermine Europe’s treaty on passport-free travel if other countries didn’t take in more of the hundreds of thousands of people arriving from the Middle East and elsewhere in search of a better and safer life.

Calling the move “a signal to Europe,” German Interior Minister Thomas de Maizière said that the government “will live up to its humanitarian responsibility. But the burden connected with the large number of refugees must be distributed in solidarity throughout Europe.”

The decision is an abrupt reversal from Germany’s decision to fling open its doors just over a week ago, when it and Austria agreed to allow in thousands of migrants being held back by authorities in Hungary. But as thousands more people cross into Germany daily, local authorities warn they are reaching their breaking point and running out of shelter space. Mr. de Maizière said the checks were aimed at curbing the flow of people into the country.

Some 40,000 migrants were expected to have arrived in Germany this past weekend alone, mainly crossing into the country from Austria—the initial focus of the border controls.