GENEVA — At least 15 countries have agreed to set up special quotas for fugitives fleeing Syria’s civil war, marking a shift in international thinking about how to deal with the world’s biggest humanitarian crisis, the head of the United Nations refugee agency said Tuesday.

The countries, including the United States and many in Europe, agreed to help resettle civilians fleeing the 30-month-old conflict, António Guterres, the high commissioner for refugees, said at a news conference. He was speaking on the sidelines of a meeting in Geneva attended by senior officials from the four countries that have absorbed most of Syria’s refugees — Lebanon, Jordan, Turkey and Iraq — and other aid-giving countries.

The offers came as Lebanon’s president, Michel Suleiman, told the United Nations General Assembly in New York that his country was being overwhelmed by the flood of refugees from Syria. His minister of social affairs, Wael Abu Faour, told diplomats in Geneva on Monday that they risked “losing a major ally” if they did not provide more support.

“Nothing of significance has materialized so far, not one hospital, not one school,” Mr. Faour said. “We are more than disappointed. We are frustrated. It has been more than two years of advice, of lessons, of promises and nothing.”