Slovakia, Romania, Czechoslovakia and Hungary voted against the plan, while Finland abstained.
The deal is intended to relieve the pressure on countries such as Greece and Italy, where the majority of migrants and refugees have been arriving.
There were hopes the emergency summit of interior ministers would bring about consensus between EU members.
Instead, member states forced through a majority vote against the wishes of the Czech Republic, Hungary, Romania and Slovakia.
The Vice President of the EU, Frans Timmermans, says all members must now accept the deal.
"We know that some member states were not in favour of the proposal. But those member states said 'we'll have a vote. You know our position, but we will respect the outcome of the vote.' And so these member states, all of them, respect the outcome of the vote. I even said afterwards what anyone voted around the table is no longer relevant once you have a decision. And this was confirmed by everyone around the table. Nobody challenged that decision."
Under the EU's constitution, a country that does not agree with a policy on migration imposed upon it has the right to appeal to the European Council.
A country can appeal if it feels the fundamental principles of its social security or legal systems are under threat.
Howeber, Luxembourg Foreign Minister Jean Asselborn, who chaired the meeting, said he had no doubt opposing countries would implement the measures.
"We have reached a deal in the council with a very large majority, a majority that considerably exceeds the requirements in the treaties. We would have preferred an adoption by consensus, but we didn't reach it. It's not for lack of effort. If I may say so, certain member states didn't join this large majority. These (countries) have put forward other views that are surely legitimate, but I have no doubt that they will put the decisions into practice."
Britain has refused to take part in the resettlement of the 120,000 and has a legal exemption.
Two countries with similar opt-outs, Ireland and Denmark, are offering to participate.
Of the 120,000 to be divided between the remaining EU states, the nine countries of central and eastern Europe are each being asked to take around 10,000, while Germany and France between them will take double that number.
However the United Nations Refugee Agency says a relocation program alone will not solve Europe's migrant crisis.
The 120,000 people to be shared across the bloc are equivalent to just 20 days' worth of arrivals at the current rate.
The UNHCR is calling for the establishment of facilities capable of receiving tens of thousands of people.
Spokeswoman Melissa Fleming has urged the European Union to establish adequate facilities, particularly in Greece.
"We believe this may be the last opportunity for a coherent European response to manage a crisis that is increasing the suffering and exploitation of refugees and migrants and it is also increasing the tension between countries."
Others have called for more to be done to address the causes of the migration crisis.
Regional director of the Norwegian Refugee Council, Carston Hansen, has told Al-Jazeera one way to solve the crisis is to help Syria's neighbours.
"We have more than 4 million refugees sitting in countries around Syria. Those refugees have been sitting there for almost 5 years now. The assistance available to them is decreasing by the day. Consequently people find themselves in more and more desperate situations and some are seeking the difficult route to Europe, the life-endangering route to Europe, in order to look for better options."
The relocated refugees will mostly be from Syria, Eritrea and Iraq.
The deal must now be ratified by EU leaders meeting in Brussels tonight.
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