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But the explosions seemed to follow them, like a slow-moving thunderstorm. So on that afternoon in July 2012, the family — Abdulrazak, Marim and their four young children — pushed on, from one neighborhood to another. Finding no respite, they got a ride across the border into Lebanon and then south to Beirut.

Their migration had only begun. Within weeks they would be on the move again, to Cuba, and eventually to Paterson, N.J., where they finally arrived last November, among the first of a small number of Syrians refugees to be resettled by the United States as a direct result of the three-year-old Syrian conflict.

"We did not plan to go to Beirut, only to leave the neighborhood until the shelling was over," said Abdulrazak, 48. "I was never thinking of going to the States."

The family is part of one of the largest refugee exoduses in recent history — more than two million Syrians have fled their country since the start of the war in 2011, and the United Nations predicts that the numbers could exceed four million by the end of this year.

With no end to the conflict in sight, pressure has been mounting on the international community to step up its efforts to help alleviate a worsening humanitarian crisis.

Abdulrazak and his family are in the vanguard of what is expected to be a small but gradually growing stream of refugees resettling in the United States this year.

Speaking through an Arabic interpreter on a recent afternoon, the family members told the story of their flight while sitting in their sparsely furnished apartment, which occupies the second floor of a well-worn clapboard-style house in Paterson.

"If we didn't fear for our lives, we wouldn't have left our country," said Marim, 37. "Our country is very precious to us." The family asked that their last name not be used, for fear that relatives still living in Syria might be targeted.

Soon after arriving in Beirut, they learned that their house had been destroyed in the bombing. "At that point, I lost hope to go home," Abdulrazak said.

The family consulted with officials at the United States Embassies in Beirut and Amman, Jordan, and with officials of several European Union nations, but they were given no hope that they would all be able to get visas.

Coincidentally, the family had been planning to take a vacation in Cuba and had already been issued tourist visas. So they decided to try their luck there. "It was a gamble," Abdulrazak said, "but it was better than Syria and Lebanon."

They flew to Havana in September 2012, and the family applied for refugee status through an office of the United Nations, Abdulrazak said. While they waited for the outcome of their petition, the children attended Spanish classes at a private school for foreigners; the United Nations paid the rent for the apartment where they stayed, he said.

They were approved for admission to the United States in October and flew to Miami on Nov. 19. The following day they flew into Newark, where they were met by relatives, who took them to their new home in Paterson, which has one of the largest Arab populations in the United States. (There are some 175,000 people of Syrian descent living in the United States, according to the Census Bureau, with about 28,000 living in the New York metropolitan area.)

"When you lose your house and your business," Abdulrazak said, "you have to be courageous."

The United Nations has set a goal of temporarily or permanently resettling at least 30,000 Syrians by the end of this year. At least 20 countries have pledged a total of at least 18,300 spaces.

The United States, which has already provided about $1.3 billion in humanitarian assistance to the Syrian people, has resettled at least 87 Syrians since fiscal year 2012, State Department officials said. The paperwork for a vast majority of those cases, however, had begun before the onset the conflict.