PUTRAJAYA, Malaysia—Indonesia and Malaysia agreed Wednesday to provide temporary shelter to thousands of migrants stranded at sea, the first breakthrough in the humanitarian crisis confronting Southeast Asia after weeks of reluctance by the region’s nations to take responsibility.

The announcement was made by Malaysian Foreign Minister Anifah Aman after a meeting with his Indonesia and Thailand counterparts to address the plight of the Rohingya Muslim and Bangladeshi migrants fleeing persecution and poverty.

“Indonesia and Malaysia agreed to continue to provide humanitarian assistance to those 7,000 irregular migrants at sea,” Mr. Anifah said, adding that the two countries “also agreed to provide them temporary shelter provided that the resettlement and repatriation process will be done in one year by the international community.”

A few thousand migrants have already made it to shore in Indonesia’s Aceh province and Malaysia’s Langkawi island.

Most of the refugees are believed to be victims of human traffickers, who recruit them in Myanmar’s Sittwe province and in Bangladesh with promises of safe passage to Malaysia, and jobs once they land there.

The three Southeast Asian countries will also take steps to bring the perpetrators of such heinous crime to justice, Mr. Anifah said.

Thailand has already arrested several police and local officials after authorities uncovered jungle camps where the trafficked Rohingya had been held until their families agreed to pay ransom for their release.

“The enforcement agencies of the countries concerned will continue to share intelligence information in their efforts to combat people smuggling and human trafficking,” Mr. Anifah said.

Mr. Anifah also urged the international community to “uphold their responsibility and urgently share the burden of providing the necessary support to Malaysia, Indonesia and Thailand in addressing the problem.