Simpson, 36, a two-time Olympic medallist, had been sailing on the Artemis, Sweden's entry in the America's Cup, when the catamaran overturned and trapped him underneath, according to a statement posted on the Artemis Racing website.
"The entire Artemis Racing team is devastated by what happened," CEO Paul Cayard said in the statement. "Our heartfelt condolences are with Andrew's wife and family."
"We obviously had a tragic day today on the bay. It's a shocking experience to go through. We have a lot to deal with for the next few days in terms of ensuring everybody's well-being," he said.
The Artemis website said Simpson was part of an 11-member sailing team and that all other crew members had been accounted for following the mishap.
Simpson won the Olympic Star class gold with Iain Percy - his Artemis team boss - in Beijing in 2008 and they took silver in London in 2012.
The twin-hulled boat was performing a so-called "bear-away" manoeuvre, turning away from the wind, when one bow dropped under the water's surface, and the vessel flipped over, America's Cup spokesman Tim Jeffery told Reuters.
Simpson ended up trapped beneath the boat in the water and had to be pulled out by rescue divers, who tried and failed to revive him, Jeffery said.
The yacht was "very badly damaged," but the team has a backup boat that is expected to be ready to sail in June, he said.
The British Olympic Association said it was "devastated" at the tragic death of Simpson.
"Andrew was a treasured and accomplished member of Team GB, both at the home Olympics in London 2012, where he won silver with childhood friend Iain Percy in the Star class, and at the Beijing 2008 Olympic Games, where he won the gold medal also sailing alongside Iain," a BOA statement read.
"Andrews's talent and humour was an inspiration to others and he will be sorely missed by the Olympic Family."
The incident was believed to be the first fatality in connection with the America's Cup since the early 1990s, when a crew member from a Spanish team died in a training accident off the coast of Majorca in the Mediterranean, Jeffery said.
San Francisco Fire Department spokeswoman Mindy Talmadge confirmed that one male crew member had died in the accident and said he was estimated to have remained under water for 10 to 15 minutes before he was recovered.
She said one other sailor from the Artemis was taken to a hospital with minor injuries and later released.
The precise number of people aboard the Artemis when it tipped over was not immediately clear.
Although America's Cup vessels sail with 11 crew members when they race, they are known to carry one or two more or one or two fewer members on practice runs.
A US Coast Guard officer, Pam Boehland, said support boats had pulled 12 crew members from the water, and that one was taken to a San Francisco hospital.
Boehland said the Artemis capsized at about 1300 local time (2000 GMT) northwest of Treasure Island. The crippled vessel remained in the bay, and the Coast Guard established a 100-yard (91-metre) safety zone around the boat, she said.
Winds on Thursday were blowing on the water at 18 to 20 knots, or about 23 to 25 miles per hour, which race organizers described as typical for the bay.
"The boats are designed to sail and compete in winds of up to 30 knots ... so this wasn't excessive," one America's Cup official told Reuters. "It was windy but not super-extreme."
Race organizers said Simpson was assigned as the crew's tactician but also would have been involved in operating winches and performing other tasks necessary to sail the twin-hulled boat.
Clive Woodward, director of elite performance for the British Olympic Association, wrote on Twitter: "Totally shocked to hear Olympic star Andrew 'Bart' Simpson died after Swedish America's cup catamaran capsizes - thoughts with his family."
John Derbyshire, Royal Yachting Association performance director, said: "We’re devastated by the news from San Francisco today. Andrew is someone I’ve worked closely with since the age of 16 – he was a great talent, and a key figure in our World Class Programme over many years culminating in his well-deserved Olympic success. He was a huge inspiration to others, both within the British Sailing Team and across the nation and our deepest sympathies go out to his family at this terrible time."
The wreck was at least the fourth major accident in just over a year off the California coast or involving California vessels.
In March, a crew member was killed when a 30-foot sailboat broke apart in rough seas during a race near San Clemente Island.
In April 2012, four crewmen in a race from Southern California to Mexico died after their yacht ran aground. Two weeks earlier, five sailors died in a racing accident near the Farallon Islands off the coast of San Francisco.
The Farallon incident prompted the Coast Guard to temporarily suspend racing in the Pacific Ocean off northern California.