Two Royal Air Force Sea King helicopters and two other support helicopters winched 79 people from the deck of the ship and from two life rafts in the water when the incident occurred on Wednesday.
A Royal Navy vessel also brought two more lifeboats with more than 200 people on board to safety.
"The swift action taken by Royal Navy and Royal Air Force crews, working with civilian counterparts, saved many lives and prevented this developing into a human tragedy," Defence Secretary Michael Fallon said.
"We responded with everything we had yesterday to assist in what was an extremely complex and hazardous rescue operation in difficult conditions," said Darren Bone, commander of British forces for the South Atlantic Islands.
Passengers and crew were taken to a British base on the Falkland Islands where they were given clothing, food and medical attention.
The ship, operated by French company Ponant, was just north of the Falklands en route to South Georgia, a British territory in the South Atlantic, when the incident began.
"A fire of a technical nature broke out on the ship Le Boreal in a compartment of the engine room," Ponant said in a statement.
"As a precaution, all passengers were evacuated from the ship," it said, adding that they were transferred to another ship from Ponant's fleet -- L'Austral -- and taken to Port Stanley in the Falklands.
"They will be directly repatriated from there by Ponant," the statement said.
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