They were among many who try to cross from the Turkish coast to this Greek island, most of them refugees from the war in Syria and the fight against Islamic State extremists. The short, but sometimes perilous, sea crossing is a crucial leg in a long journey that many hope will lead them to asylum in Northern Europe.

Some 165,000 Syrians have sought safety in the European Union since the start of the war more than three years ago, according to Eurostat, the EU statistics service. In warm-weather months when the water is calm, those attempting the crossing into Greece have numbered in the hundreds, in a daily drama that has left rescuers scrambling to respond.

Smugglers who ferry the migrants prefer the route to Chios because it is one of the fastest between Turkey and the EU. In some places, the distance is as short as 5 miles (8 kilometers) and takes a half-hour in a small inflatable boat with a modest single engine.

As Syria's crisis has escalated, smuggling operations have grown more sophisticated and refugees in camps in Turkey, Jordan and Lebanon are increasingly desperate to get out. In the nine months to September, 4,552 migrants were rescued in the waters around Chios alone—more than twice the number for the whole of 2013. In 2012, the figure was 69, according to the Greek police.

The influx is a major political issue in the EU, with Greece and Italy overwhelmed by the numbers of people trying to reach their shores and the costs of rescuing them. Wealthier Germany and Sweden—where many Syrians end up—argue their asylum systems are overstretched.

Chios, one of the bigger Greek islands with a population of some 53,000, and neighboring islands, Lesbos and Samos, are the main entry points to Greece for refugees, the overwhelming majority Syrian. From January through September this year, almost 19,000 arrived on the three islands, 40% of the country's total. There aren't clear numbers on how many have died during the journey, though it is safer than other routes.

As one of the boats approached Mr. Khalif and his friends, he broke into a big smile and raised his hand out of the water to make the victory sign. Crew members hefted the three aboard with a rope.

Collapsing on the rescue boat's deck, Mr. Khalif still couldn't believe he was in Greece. Now in the EU, he can file a claim for asylum and hope for a new life away from the bombs in Deraa, his hometown, or the destitution of Turkish refugee camps.

"Yunan, Yunan?" Mr. Khalif asked between short breaths—Arabic for "Greek." One of the coast guards showed him the Greek national insignia on his uniform shirt.

Mr. Khalif brought his palms to his face, bowed his head for a moment, then raised his eyes to the heavens in prayer.

Sometimes migrants risk all to ensure they are rescued and processed in the EU. That night, 35 migrants aboard a small inflatable boat tore it up with a sharp object just as a coast-guard vessel approached. It began to sink as the officers rushed to them, leaving some thrashing in the water and pleading for help.

Refugees often tear their inflatable boats to make sure the coast guard rescues them.

Crew members threw them a rope, barking at them to stay calm as people rushed to grab it. "Sir, sir, I can't swim," a young man shouted. A middle-aged man struggled to hold on to his glasses as he tried to stay above water.

Once on board, one migrant said smugglers in Turkey had instructed them to sink the boat to ensure authorities are forced to save them. The Greek coast guard said this is common practice.

But sometimes migrants are so suspicious the Greek authorities will send them back to Turkey that they refuse to stop before making land. Human-rights groups and the United Nations High Commissioner for Refugees have repeatedly said they believe the Greek coast guard pushes refugees back to Turkey—illegal under international law. Greece denies its officers do this and has launched investigations into a handful of alleged incidents.

Long-strained relations between Greece and Turkey make it harder to manage the influx. Greece and Turkey patrol this sea border with armed navy vessels, vigilant for territorial violations from the other side. Turkey has made claims on Greece's waters in the eastern Aegean Sea, which Greece rejects, in a decadeslong diplomatic saga that in 2006 nearly led the two North Atlantic Treaty Organization allies to war.

The day after the inflatable boat sank, a Greek coast-guard vessel was floating on the unmarked water border. The lights on Chios and the Turkish town of Çeşme were so near, and the waters so calm, it felt as if the boat was in the middle of a small lake.

The Greek captain alerted his Turkish counterpart, patrolling nearby, that he could see 50 people gathering on the Turkish coast on his thermal radar. What was meant as an early warning turned into a 20-minute exchange over whether the Greek coast guard was in Turkish waters or not.

The Turkish coast guard said it has caught almost 11,000 migrants trying to cross to Greece this year, three times the number in 2012. As many as 500 were apprehended in a single week in late September. Several refugees on Chios said they had been taken back to Turkey by the authorities during previous arrests, but were let free and attempted to cross again.

Migrants drive their boat toward the coast, fearing they might be turned back by the Greek coast guard.

The migrants on the coast got into their dinghy and were rescued off Chios later that morning by the Greek coast guard. From there they were taken to a makeshift reception camp, a parking lot by the port-authority building at the edge of the harbor, featuring three tiny wooden huts, a small shipping container converted into a place to sleep, two chemical toilets and two dusty doorless tents, filled with pallets and dirty foam mattresses.

Among the group were 31-year-old Muhammad Husein and his wife, Asil, 25. The two are Syria-born Palestinian refugees and grew up next door to each other in the Palestinian refugee camp of Khan Eshieh, near Damascus.

Mr. Husein, a finance major, left Syria for Dubai in 2008 where he worked as a banker for Standard Chartered PLC. Mrs. Husein stayed in Syria and graduated two years ago as a food engineer, but Dubai wouldn't accept her as a resident. "I couldn't leave her in Syria," he said, holding her hand.

While Mr. Husein was working, his wife survived bombings and saw their old neighborhood partly flattened. Both their families fled to Lebanon, but the couple decided that with their educations they would have a better chance in Europe.

The wall street journal