Under the deal, migrants arriving in Greece are now expected to be sent back to Turkey if they do not apply for asylum or their claim is rejected.
The influx of people crossing to Greek islands grew ahead of the deadline, and Greece said it would not be able to implement the deal immediately.
On Sunday, volunteers on the Greek island of Lesbos were dealing with the first arrivals under the new regime.
Just hours before the agreement came into force, a four-month-old baby girl drowned when a boat carrying migrants sank off the Turkish coast, Turkey's Anadolu agency reported.
The deal says that for every Syrian migrant sent back to Turkey, one Syrian already in Turkey will be resettled in the EU.
However, there were still many doubts about the implementation of the agreement, including how the migrants would be sent back.
Some 2,300 experts, including security and migration officials and translators, are set to arrive in Greece to help enforce the deal.
But Greek officials said none of the experts had yet arrived and the deal could not be implemented immediately as key details still needed to be worked out.
"A plan like this cannot be put in place in only 24 hours," said government migration spokesman Giorgos Kyritsis, quoted by AFP.
With the deal, it is hoped people will be discouraged from making the dangerous journey by sea from Turkey to Greece. In return, Turkey will receive aid and political concessions.
Since January 2015, one million migrants and refugees have entered the EU by boat from Turkey to Greece. More than 143,000 have arrived this year alone, and about 460 have died, according to the International Organization for Migration.
The Turkey-EU statement in full
Most of them are keen to go to Germany and other northern European Union countries, and tens of thousands are now stuck in Greece as their route north has been blocked.
Critics, however, have said the deal could force migrants determined to reach Europe to start using other and potentially more dangerous routes, such as the journey between North Africa and Italy.
On Saturday, the Italian coastguard said more than 900 people were rescued amid an increase in traffic through the Strait of Sicily.
And Libyan authorities said the bodies of four women were recovered but at least 20 others were still missing after a boat carrying migrants sank off the country's coast.
Officials there said they rescued nearly 600 people from three other boats on Saturday.
Human rights groups have strongly criticised the deal, with Amnesty International accusing the EU of turning "its back on a global refugee crisis".
On Saturday, thousands of people protested in support of refugees and against racism. Rallies were held in London, Athens, Barcelona, Amsterdam, Geneva and some other cities.
In the Greek capital, protesters, including some Afghan refugees, chanted "Open the borders" and "We are human beings, we have rights".
In London, about 4,000 people joined a protest carrying placards with slogans like "Refugees welcome here" and "Stand up to racism".
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.

This article was about:
An EU-Turkey deal to tackle the migrant crisis has formally come into effect.