The European Union on Sunday agreed with Turkey’s government for Ankara to take steps to cut the flow of migrants into Europe in exchange for EU cash and help with its bid to join the 28-nation bloc.

EU leaders hailed the agreement as a key step toward substantially reducing the number of asylum seekers entering the bloc, while Turkey’s Prime Minister Ahmet Davutoglu said Sunday’s summit marked a historic new beginning in the often fraught relations between Brussels and Ankara.

Yet...