The mother flagged down the American search-and-rescue team on Monday in the tropical heat, one desperate figure among thousands in the aftermath of Venezuela’s catastrophic earthquakes.

Her three children, ranging in age from 9 to 16, had been in an apartment on the ninth floor of a 12-story building that collapsed. Parts of her town, Caraballeda, in the hard-hit state of La Guaira, were leveled.

The American crew had actually been on its way to a different disaster site. But it stopped, said Trey Espy, an assistant chief with the Los Angeles County Fire Department and the crew’s task force leader. Two search dogs signaled that someone below the mountain of rubble was alive. Special listening devices detected the sound of tapping. That’s when the crew called its base camp for special excavation equipment.

Since then, workers have been digging — with concrete cutters, shovels, axes and bare hands.

“We know time is ticking,” he said.

The team members from Los Angeles, along with scores of specialists from Fairfax County, Va., and Miami-Dade County, Fla., are among workers from around the world who have joined the frantic effort to save who they can.

The back-to-back earthquakes that devastated large coastal areas near Caracas, the Venezuelan capital, have killed at least 1,900 people. Officials believed thousands more were still trapped in hundreds of collapsed structures.

Speaking by phone from Venezuela, Mr. Espy described in bleak terms what he and other members of his team were seeing. “Buildings are crumbled,” he said. “There’s no place to get gas. No place to buy groceries. No running water or electricity. It rained pretty heavily last night, and people couldn’t get shelter.” Some people, he added, “have been entombed for almost a week without food or water.”

Working together, the teams from Los Angeles and Fairfax Counties, which arrived on Friday — about 48 hours after the quakes — have rescued five people from the rubble in La Guaira, according to John Morrison, a spokesman for the Fairfax County Urban Search and Rescue unit, who also is with the crew in Venezuela.

It is the type of mission they prepared for. Just last month, about 80 members of the Los Angeles County unit traveled to Virginia for a joint exercise with their counterparts from Fairfax County. The two units, which are often deployed by the U.S. government when natural disasters strike foreign countries, spent five days working through how they would respond to a large-scale earthquake.

The teams, which traveled to Venezuela on separate military cargo planes that each hauled about 80,000 pounds of equipment, have set up camp and expect to be in Venezuela for about two weeks, Mr. Morrison said. Another urban search-and-rescue crew, from Miami-Dade County, is working separately, but all three units are being directed by the State Department.

Mr. Morrison said he had witnessed three rescues so far, and described them as painstakingly deliberate, taking anywhere from eight to 14 hours. The units have often relied on local emergency workers, relatives of family members who may be trapped and people who live in the community to let them know where there might be a sign of life — a voice or a knocking sound. Rescue dogs are sent to sniff out the scent of live or dead bodies from the rubble pile.

If there’s a sign of life, Mr. Morrison said, a listening device, which can detect water dripping, a rat scurrying or a curtain ruffling against a wall, is dropped into a space. If rescuers hear a voice, they triangulate where it is loudest and try to drill holes in the concrete for a camera that can give a picture of the maze.

Structural engineers on the team are tasked with figuring out if the rubble can bear a load and how to move it without harming the survivor or the rescue workers. “There’s a lot of breaching and breaking,” Mr. Morrison said.

All that came into play in the early Sunday morning darkness.

There were signs that a 58-year-old woman lay in the rubble of an eight-story apartment building that had collapsed. After eight hours, rescue workers had navigated a path close enough that they could drill a one-foot hole in a wall that gave them access to the woman’s foot.

That allowed Dr. Natalie Sullivan, an emergency medicine physician at George Washington University Hospital, to climb through the wreckage to administer an IV with fluids and medication. When workers removed more debris, a straw was inserted into the pile and the woman was able to drink water.

Dr. Sullivan, who speaks Spanish, said the woman told her that she was meeting with a prayer group by the apartment building’s swimming pool when the ground began to shake. The woman told her that she ran into a stairwell when the building collapsed.

“She’s a very brave woman,” Dr. Sullivan said.

The intravenous fluids are essential, Dr. Sullivan said, to prevent crush syndrome, the release of toxins into the bloodstream after heavy debris is lifted from someone’s body after a lengthy period. It can lead to kidney failure and cardiac arrest, she said.

“Once we’re able to move them, they can get really sick really fast,” Dr. Sullivan said.

The woman, dehydrated and with an apparent broken rib, was carried out from the debris about 12 hours after the workers arrived. The teams also rescued a mother and a 9-month-old baby, and on Sunday rescued a father and son.

As the local emergency workers and the Americans have carried survivors out hand-in-hand, Mr. Morrison said there is a bond that transcends language or cultural barriers. “It’s high-fives all around and then, ‘Let’s get back to work and find more survivors,’” he said.

Even for Mr. Morrison and Mr. Espy, who have been on similar missions in places like Nepal and Turkey, the scale of the destruction and the death toll has been sobering, they said.

But in those rescues of a handful of survivors, the rescue workers have found a measure of solace.

“When a 12-story building collapses, it’s a miracle if you survive,” Mr. Espy said. “But miracles are still happening.”

On Tuesday afternoon, at the mountain of rubble where they heard tapping, they were still digging.