The 40-foot boat known as the Sea the City 2, which allows passengers to relax in a pair of hot tubs, departed the dock at Liberty Landing Marina at around 10 p.m. Saturday with 17 young passengers. The occasion was a birthday, according to the boat’s captain, Josh Cochran, and the plan was to cruise for an hour and a half.

New York Harbor was fairly calm as they headed out from a nearby marina. Mr. Cochran piloted the boat by the Statue of Liberty and was headed toward the Brooklyn Bridge when he spotted a tiny slash of light moving in the water.

“It’s not uncommon to see flashing lights in the water,” said Mr. Cochran on Tuesday in a telephone interview. “It’s usually a glow stick or party favor.”

The festive hot tub cruise soon turned into a rescue mission, as the Sea the City 2 found 12 people clinging to a capsized vessel.

Mr. Cochran’s account fills out new details of a harrowing night in which a mother and her daughter lost their lives in an incident on the water that has triggered a federal investigation and led to the arrest of the pilot of the capsized boat. Law enforcement officials confirmed the sequence of events as outlined by Mr. Cochran.

The captain of the hot tub boat said he had not heard anything about a vessel in distress on his marine radio, but decided to check on the light in the water. He ordered a crew member, Nicole Ketchledge, and another captain, Scott Krute, to the bow as lookouts.

Just ahead they saw a capsized vessel with a dozen people hanging onto sides.

“They were very much in distress,” Mr. Cochran said.

In an eerie symmetry, the overturned vessel had a similar name to the boat that now approached it: the City on the Sea. Like the Sea the City 2, the smaller vessel was also a charter boat taking passengers out in the harbor to celebrate a birthday. And it had also just passed the Statue of Liberty.

The City on the Sea was a 21-foot pleasure boat that had set off with 13 people plus a captain — too many people on a boat with a seating capacity of 10, New York City police officials would later say. It had come down the East River to the harbor, then flipped after hitting the wake likely created by a larger vessel, according to court documents.

The guests were now stranded in the middle of the channel, drifting southward.

The boat was in a busy section of the harbor used during the daytime by the Staten Island Ferry, barges and other maritime traffic, Mr. Cochran said. “But that time of night — it’s pretty sparse traffic out there,” he said.

Mr. Cochran said he turned the music off, radioed for help on his marine radio.

“We’re not a fast vessel, so we’re not accustomed to responding to these kind of things,” he said.

Still, he ordered his crew to grab emergency equipment, including life jackets, a life ring and boat hook typically for lines. A deck hand, Caitlin Brunnock, grabbed a rescue ladder to help the people on board.

Passengers from the capsized vessel began climbing up the ladder on to the hot tub boat. Others clambered over the side. “People just climbed up any way they could,” Mr. Cochran said.

“You had 12 panicked people yelling for help, and we’re yelling directions to each other, how to get them on the boat,” Mr. Cochran said. Even his passengers in the hot tubs offered to help, he said, “But we said, ‘Just stay in the tubs for now,’” he said. “We didn’t need 17 more people running around the boat.”

Some guests aboard the larger party boat used the flashlights from their phones to help illuminate the scene. “They did absolutely everything they could to help,” the Mr. Cochran said.

Mr. Cochran said he heard people from City on the Sea yelling about how two from their party were missing, “But we weren’t able to see where they were at,” the captain said.

The missing members of the party would later be identified as Sara Sánchez, 27, and her 5-month-old daughter, Antonella. They were later found unresponsive in the water by police divers, and pronounced dead shortly afterward.

By the time the Sea the City 2 came to their rescue, the people from the capsized boat said they had been in the water about 20 minutes. “Some said they were trying to call 911,” he said, “but their phones were too wet to work.” The crew began handing out bottled water, along with towels they typically rent to the hot tubbers, said Mr. Cochran.

“One of our crew overheard a boat guest say, ‘It was a wave, we hit a wave,’” Mr. Cochran said. “I would guess, if anything, it would have been a large boat wake.”

The baby’s father, Andrés García, remained in a panic and did not want to leave the capsized boat.

“He kept saying, ‘My daughter my daughter.’” He was very much hysterical,” Mr. Cochran said. “He kept saying, ‘You have to help, we have to find her.’ But we didn’t have the means.”

Around 10:30, the first rescue boats arrived, Mr. Cochran said.

Minutes later, a helicopter rescue unit arrived at the scene with police divers. Two Coast Guard boats, two boats from the Police Department and four Fire Department vessels also joined the rescue effort.

Mr. Cochran said the shaky rescued passengers, some seemingly in shock, were moved onto marine rescue vessels.

Mr. García remained in a panic and did not want to leave the capsized boat.

“He was saying, ‘You got to help me find my wife and daughter,’” Mr. Cochran said. “They said, ‘We’re doing the best we can.’ He was emphatic that he wanted to go with them and help and they had to tell him, ‘No, you need to stay here.’”

Manuel Hernandez, 46, the boat’s captain, was later arrested and charged in federal court on Monday with two counts of misconduct and neglect resulting in two deaths. He and his court-appointed lawyer declined to comment on Monday as he left federal court. He did not answer phone calls or messages left at his home on Tuesday.

By the time Mr. Cochran took his own passengers back to Liberty Landing Marina, it was almost midnight.

“Everyone was emotionally spent,” he said. “We said, ‘We’re sorry it had to happen when you were here, but they were very grateful to the crew. They said, ‘Good job out there.”

Davaughnia Wilson, Miles G. Cohen and Mark Bonamo contributed reporting.